Collection of Infinite Products and Series

   Dr. Andreas Dieckmann, Physikalisches Institut der Uni Bonn

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My interest in infinite products has its origin in the year 2000 in connection with the problem of the electrical field of a
line charge trapped inside a rectangular tube. After I learned that the double product can be solved using
elliptic theta functions I was hooked. The site has been growing ever since, and its focus has been expanded
to include Series as well.

These pages list thousands of expressions like products, sums, relations and limits shown in the following sections:

-  Infinite Products

-  Products involving Theta Functions

-  Other formulae and curiosities including sums of hyperbolic and inverse tangent (arctan) functions and q - series

-  q-Series

-  special values of EllipticK and EllipticE

-  Series of Hyperbolic Functions

-  Series of CosIntegral

-  Sums involving reciprocal multifactorials or factorials

-  Multiple Sums (lattice sums)

-  diverse Series

-  Series of Logarithms

-  Series of Inverse Tangents ( Arcustangent )

-  Series of Bessel Functions

-  Series of Legendre Polynomials

-  Series of Zeta PolyGamma PolyLog and related

-  Series of Beta Functions

-  Series of Gamma Functions

-  Series involving HarmonicNumber

-  Series involving Hypergeometric Functions

-  some Limits

-  a few Integrals

-  iterated expressions ( Tetration )

-  some properties of ProductLog LerchPhi and PolyLog

{j, n, m} are Integer; {λ, q} > 0 and r are real; {z, InfProd_3.png, InfProd_4.png, InfProd_5.png, InfProd_6.png} may be complex; Γ[a] is  Gamma[a];
InfProd_7.png], InfProd_8.png] are shorthands for the Elliptic Integrals,
sl[x]  cl[x] and ϖ denote the lemniscate functions and constant InfProd_9.png;
the notation Σ’ means that the divergent term in multiple sums is excluded.
There are  products that possess pointlike poles, where the denominator of a factor gets zero for certain
values of z. The given domains may not be complete. Some of the expressions are well known,
others may be not; some were found in the depths of the world wide web, the first are derived
from the first product below.

any formula you decide to use should be numerically tested for validity in the users domain  ←

Expressions communicated by other people are marked with (R#) and are referenced below at the bottom.

Infinite Products : ( Back to Top )

InfProd_10.png

This product converges and delivers infinite product representations for many functions if the {a, b, c, d} are
replaced by constants and simple functions of z :

InfProd_11.png

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InfProd_14.png

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Products of two Gammas :

InfProd_61.png

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Partial Fraction Decompositions :

InfProd_64.png

General expression :

InfProd_65.png

some special cases (all having m = 1, except where noted otherwise):

Order 2:

with n = 2 :

InfProd_66.png

most common  case (a quadratic binomial)

with  InfProd_67.png  and  InfProd_68.png:

InfProd_69.png

Decomposition of the general quadratic trinomial applying the shorthands

InfProd_70.png

gives

InfProd_71.png

Order 3:

with n = 3  :

InfProd_72.png

With 3 abbreviations

InfProd_73.png

the general cubic multinomial  can be written in terms of first order in x:

InfProd_74.png

Order 4:

A simple one :

InfProd_75.png

And at last using these 9  subexpressions

InfProd_76.gif

the general multinomial  of 4th order (n = 4) looks  like (expressed again in terms of  first order in x):

InfProd_77.png

  InfProd_78.png as simple function of k :

Decomposition of  finite products into power series

InfProd_79.png

The general case (r determines the start index of the product, the coefficients of x are called r - Stirling Numbers of the first Kind e.g.Oeis:A143493) :

InfProd_80.png

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Special cases with m = 0 :

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q - Product (0 < q < 1) :

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Two kinds of decomposition of the same product :

InfProd_95.gif

The form of the original product returns in the coefficients of the second decomposition sum, so there is a recurring structure, like :

InfProd_96.png

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(left hand side taken from the general expression for partial fractions above) equations like given for instance in the next line are valid for different functions f[k] :

InfProd_98.png

If f[k] = InfProd_99.png  then (0 < x)

InfProd_100.gif

or if f[k] = InfProd_101.png  then

InfProd_102.png

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InfProd_104.png

InfProd_105.png

InfProd_106.png

More Products :

The maximum of the next function is found at InfProd_107.png | f InfProd_108.png)) =  InfProd_109.png,  its zeros on the positive (negative) axis are the odd (even) Integers and in general f[z] = f[-z-1] :

InfProd_110.png

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Euler’s product :

InfProd_132.png

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The idea for the following product is taken from: Symmetry 2022, 14, 1418. https://doi.org/10.3390/sym14071418 .

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The next product approximates a Gauss function InfProd_142.png with standard deviation InfProd_143.png for a large value of m >> n:

InfProd_144.png

The agreement of the above approximation for m  > InfProd_145.png increases with j and the error is smaller than InfProd_146.png in the intervall (0 < n < 4InfProd_147.png) .

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Products of trig functions :

InfProd_155.png

Vieta' s product (set x → π/2 in the first product of Cos above)  was maybe the first (1593) documented infinite product.
It was obtained by taking the ratio of the area of the square inscribed  in a circle of radius r to the area of a InfProd_156.png-polygon (built from InfProd_157.png isosceles triangles) inscribed into the same circle
InfProd_158.png,
beginning with n = 2. It ends up at n = ∞, where the area of the polygon is equal to the area of the circle with
InfProd_159.png.

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Vieta’s product can be rewritten by pulling the factors 1/2 inside the following square root factors :

InfProd_161.png

and may then be compared to its lemniscatic cousin:

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Products containing factors built from all primes p:

InfProd_208.png

InfProd_209.png

InfProd_210.png

Products involving Theta Functions    ( Back to Top )

InfProd_211.png is shorthand for EllipticTheta[n, z, q] and InfProd_212.png means EllipticThetaPrime[m, z, q].

InfProd_213.png

Series and Product Representations :

InfProd_214.png

InfProd_215.png

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With  InfProd_219.png[ 0 , q ] a few relations between the theta functions are

InfProd_220.gif

InfProd_221.png

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These limits give "needles" of height ±1 situated at the extreme values of Cos or Sin respectively (n∼1/(4λ)) :

InfProd_229.png

Approximation of quotients (0.4 < λ) :

InfProd_230.gif

InfProd_231.png

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Partial differential equation (pde) :

InfProd_233.png

EllipticThetas with imaginary argument :

InfProd_234.png

With z → 0 this reduces to

InfProd_235.png

Half Lambda :

InfProd_236.png

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InfProd_240.png

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Double Lambda :

InfProd_242.gif

InfProd_243.png

Other relations :

InfProd_244.png

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From an equation involving Eisenstein series InfProd_246.png and InfProd_247.png InfProd_248.png and their connection to theta functions:

InfProd_249.png

Square and square root of q :

InfProd_250.png

InfProd_251.png

Double Argument  (Landen), -  see above for double  InfProd_252.png)  and half  InfProd_253.pngLambda - :

InfProd_254.png

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InfProd_257.png

Half Argument :

InfProd_258.png

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Proportionality of imaginary and real part of EllipticTheta2 and EllipticThetaPrime1 with real argument  and imaginary nome 0 ≤ q :

InfProd_262.png

Derivatives with respect to q :

InfProd_263.png

InfProd_264.png

InfProd_265.png

InfProd_266.png

With help of the pde above is (second or third derivative with respect to the argument z at z = 0) :

InfProd_267.png

For  integrals of the elliptic theta functions scroll to the bottom of :

Table of Definite Integrals

Series of theta functions :

InfProd_270.png

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InfProd_272.png

Now InfProd_273.png may be extracted out of the sum because of its periodicity and the remaining sum can be done :

InfProd_274.png

InfProd_275.png

After letting q = InfProd_276.png and some algebra this may be generalized to :

InfProd_277.gif

Infinite sums of elliptic theta functions multiplied with some function f[k] depending on k (as the theta functions are periodic, they may be - up to a sign - be drawn out of the sum) :

InfProd_278.png

Series representation of ratios of theta functions :

InfProd_279.png

InfProd_280.png

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The following double products numerically converge best if k ≫ n.

InfProd_282.png

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Double product representation of the single theta functions :

InfProd_284.png

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InfProd_286.png

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If the product over k is carried out first then products remain containing Tanh or Coth :

InfProd_288.png

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The theta functions may be expressed through each other :

InfProd_296.png

and exhibit a kind of double periodicity ({m, n} ∈ Z) :

InfProd_297.png

Products with factors made up of powers or square roots of ratios of integers :

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The next product featuring the lemniscatic constant appears in a similar form compared to the Wallis product directly above (isn’t that amazing ?) :

InfProd_315.png

Both ((*1*) and (*2*)) describe a certain property of members of the family of clover curves given by the polar equation

InfProd_316.png

where r and θ denote the polar coordinates of a point on the curve  (m = 1: Cardioid, m = 2: circle, m = 3: three-leaf-clover, m = 4: lemniscate, m = 5: five-leaf-clover…).
Their principal parts are located inside a cone of width θ = {-π/m, π/m} while the respective arclengths (at r = 1) of the positive half a ‘clover leaf’ can be nicely expressed in form of a Wallis - type product as:

InfProd_317.png

For m up to 5 the arclengths of the half leaves are explicitly:

InfProd_318.png

The quotient of (*2*)/(*1*) is :

InfProd_319.png

and 1/2 of  (*1*)/(*2*), starting with k = 0, gives :

InfProd_320.png

The next Wallis - type product describes the area under a superellipse expresssed by  InfProd_321.png with a = b = 1 inside the first quadrant:

InfProd_322.png

A few values of the product above for integer s from 1 to 6 are

InfProd_323.png

where InfProd_324.png is the real half - period for the Weierstrass elliptic function with invariants g2 = 0, g3 = 1.

InfProd_325.png

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InfProd_328.gif

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Trigonometric and hyperbolic Products :

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With  m = InverseEllipticNomeQ[Exp[-π λ]] and K[m] = EllipticK[m] :   

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q - Products :

In the following is ( 0 < q < 1 ) and InfProd_374.png[ 0 , q ] ,   (InfProd_375.png[ 0 , q ] =InfProd_376.png[ 0 , - q ] ) :

InfProd_377.png

InfProd_378.png

m = InverseEllipticNomeQ[q] and K[m] = EllipticK[InverseEllipticNomeQ[q]].

InfProd_379.png

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m = InverseEllipticNomeQ[q], K[m] = EllipticK[InverseEllipticNomeQ[q]] and E[m] = EllipticE[InverseEllipticNomeQ[q]]:

InfProd_413.png

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InfProd_415.png

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InverseEllipticNomeQ m[q], K[m[q]] and E[m[q]] expressed through infinite products or theta functions:

InfProd_417.gif

InfProd_418.png

InfProd_419.png

InfProd_420.png

InfProd_421.pngInfProd_422.png and InfProd_423.png can be expressed through m[q] , K[m[q]] and E[m[q]] :

InfProd_424.png

and similarly :

InfProd_425.png

and :

InfProd_426.png

and from combining the above like :

InfProd_427.png

we get :

InfProd_428.png

as q is getting larger than InfProd_429.png the branch cut of K and E is crossed, so the continuous and smooth complex functions are built from two parts :

InfProd_430.png

or turned the other way round :

InfProd_431.png

InfProd_432.png

If the result of the imaginary transformation doesn't seem right, consider the following points :
• If in the resulting formula a sign change of the imaginary part as function of q occurs under a square root ( at q = Exp[- π / 2] ) then the square root may take the other sign
• Logs with complex arguments may end up on a wrong branch, try replacing Log[...] with Log[...] + n 2 π i

Theta Functions (z = 0, π/4, π/2, 3π/4) expressed through EllipticK and m :

InfProd_433.png InfProd_434.png InfProd_435.png InfProd_436.png
InfProd_437.png InfProd_438.png InfProd_439.png InfProd_440.png
InfProd_441.png InfProd_442.png InfProd_443.png InfProd_444.png
InfProd_445.png InfProd_446.png InfProd_447.png InfProd_448.png
InfProd_449.png InfProd_450.png InfProd_451.png InfProd_452.png
InfProd_453.png InfProd_454.png InfProd_455.png InfProd_456.png
InfProd_457.png InfProd_458.png InfProd_459.png InfProd_460.png
InfProd_461.png InfProd_462.png InfProd_463.png InfProd_464.png

Series expansion of InverseEllipticNomeQ :

InfProd_465.png

With nome q = InfProd_466.png the complementary nome is q’ InfProd_467.png= InfProd_468.png. The following development shows convergence for InfProd_469.png < q, because q’ is rapidly approaching zero with increasing q < 1 :

InfProd_470.png

Connection of InverseEllipticNomeQ to ModularLambda :

InfProd_471.png

InfProd_472.pngInfProd_473.png (n) in Wikipedia (see ' modular lambda function') .

InfProd_474.png

Special values of InfProd_475.png (n) :

InfProd_476.png

Specific Values :

InfProd_477.png

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InfProd_486.gif

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A special (remarkable) relation :

InfProd_495.png

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InfProd_497.png

InfProd_498.png

InfProd_499.png

EllipticNomeQ :

Series expansions :

InfProd_500.png

InfProd_501.png

Square and square root of the nomen :

InfProd_502.png

Specific Values :

q[m[#]] = #; from specific values of InverseEllipticNomeQ above like for example q[m[InfProd_503.png]] = q[InfProd_504.png]  =InfProd_505.png;

Ramanujans g functions:

InfProd_506.png and InfProd_507.png are Ramanujans g functions, m = InverseEllipticNomeQ[InfProd_508.png]  (for each n ∈ Integer  the even g and the odd G seem to show a somewhat simpler structure than their counterparts) :

InfProd_509.png

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products with q = InfProd_522.png :

m = InverseEllipticNomeQ[InfProd_523.png] :

InfProd_524.png

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special cases :

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Theta Functions, specific values :

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InfProd_560.gif

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InfProd_587.gif

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Beauty meets well-tempered music…;-) ↓

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InfProd_597.png

InfProd_598.png

InfProd_599.png

InfProd_600.png

InfProd_601.png

InfProd_602.png

InfProd_603.png

InfProd_604.png

InfProd_605.png

InfProd_606.png

InfProd_607.png

InfProd_608.png

InfProd_609.png

InfProd_610.png

InfProd_611.png

Other formulae and curiosities including sums of hyperbolic and inverse tangent (arctan) functions and q - series:   ( Back to Top )

The following double sums numerically converge best if k ≫ n. For numerical checks the finite lower limit should be increased by 1 in case the running index is shifted by -1/2.
ϑ ‘’ represents the second derivative of ϑ (x,q) with respect to x, eg: InfProd_612.png means InfProd_613.png.

The double series converge best numerically if k >> n.

InfProd_614.png

InfProd_615.png

InfProd_616.png

InfProd_617.png

InfProd_618.png

InfProd_619.png

InfProd_620.png

InfProd_621.png

InfProd_622.png

InfProd_623.png

InfProd_624.png

InfProd_625.png

InfProd_626.png

InfProd_627.png

InfProd_628.png

InfProd_629.png

InfProd_630.png

More double sums can be found below at 'multiple Sums (Lattice sums)'.

InfProd_631.png

InfProd_632.png

InfProd_633.png

InfProd_634.png

InfProd_635.png

InfProd_636.png

InfProd_637.png

InfProd_638.png

InfProd_639.png

InfProd_640.png

InfProd_641.png

InfProd_642.png

InfProd_643.png

Series involving exponentials :

InfProd_644.png

InfProd_645.png

InfProd_646.png

InfProd_647.png

InfProd_648.png

InfProd_649.png

InfProd_650.png

InfProd_651.png

InfProd_652.png

InfProd_653.png

InfProd_654.png

InfProd_655.png

InfProd_656.png

InfProd_657.png

InfProd_658.png

InfProd_659.gif

InfProd_660.png

InfProd_661.png

InfProd_662.png

Theta functions as series of shifted Gauss functions with the same width (aren’t these quite remarkable relations ? See below in series of hyberbolic functions for a similar phenomenon with shifted Sech and Csch functions connected to lemniscate functions) :

InfProd_663.png

InfProd_664.png

InfProd_665.png

InfProd_666.png

InfProd_667.png

InfProd_668.png

InfProd_669.png

InfProd_670.png

InfProd_671.png

InfProd_672.png

InfProd_673.png

InfProd_674.png

InfProd_675.png

InfProd_676.png

InfProd_677.png

InfProd_678.png

InfProd_679.png

InfProd_680.png

InfProd_681.png

InfProd_682.png

InfProd_683.png

InfProd_684.png

InfProd_685.png

InfProd_686.png

InfProd_687.png

InfProd_688.png

Series involving InfProd_689.png :

InfProd_690.gif

InfProd_691.png

InfProd_692.png

InfProd_693.gif

InfProd_694.png

InfProd_695.png

InfProd_696.png

InfProd_697.png

InfProd_698.png

InfProd_699.png

InfProd_700.png

InfProd_701.png

InfProd_702.png

InfProd_703.png

InfProd_704.png

InfProd_705.png

InfProd_706.png

InfProd_707.png

InfProd_708.png

InfProd_709.png

InfProd_710.png

InfProd_711.png

InfProd_712.png

InfProd_713.png

InfProd_714.png

InfProd_715.png

InfProd_716.png

InfProd_717.png

InfProd_718.png

InfProd_719.png

InfProd_720.gif

InfProd_721.png

InfProd_722.gif

InfProd_723.png

InfProd_724.png

InfProd_725.png

Series of Stirling numbers :

InfProd_726.png

InfProd_727.png

InfProd_728.png

InfProd_729.png

By the Inversion Theorem for Stirling numbers (S1 ⇔ S2):

InfProd_730.png

InfProd_731.png

Series of trigonometric functions :

InfProd_732.png

InfProd_733.png

InfProd_734.png

InfProd_735.png

InfProd_736.png

InfProd_737.png

InfProd_738.png

InfProd_739.png

InfProd_740.png

InfProd_741.png

InfProd_742.png

InfProd_743.png

InfProd_744.png

InfProd_745.png

InfProd_746.png

InfProd_747.png

InfProd_748.png

InfProd_749.png

InfProd_750.png

InfProd_751.png

InfProd_752.png

InfProd_753.png

InfProd_754.gif

InfProd_755.png

InfProd_756.png

InfProd_757.png

InfProd_758.png

InfProd_759.png

InfProd_760.png

InfProd_761.png

InfProd_762.png

InfProd_763.png

InfProd_764.png

InfProd_765.png

InfProd_766.png

InfProd_767.png

InfProd_768.png

InfProd_769.png

InfProd_770.png

InfProd_771.png

InfProd_772.png

InfProd_773.png

InfProd_774.png

InfProd_775.png

InfProd_776.png

InfProd_777.png

InfProd_778.png

InfProd_779.png

InfProd_780.png

InfProd_781.png

InfProd_782.png

InfProd_783.png

InfProd_784.png

InfProd_785.png

InfProd_786.png

InfProd_787.png

InfProd_788.png

InfProd_789.png

InfProd_790.png

InfProd_791.png

InfProd_792.png

InfProd_793.png

InfProd_794.png

InfProd_795.png

InfProd_796.png

InfProd_797.png

InfProd_798.png

InfProd_799.png

InfProd_800.png

InfProd_801.png

InfProd_802.png

InfProd_803.png

InfProd_804.png

InfProd_805.png

InfProd_806.png

InfProd_807.png

InfProd_808.png

InfProd_809.png

InfProd_810.png

InfProd_811.png

InfProd_812.png

InfProd_813.png

InfProd_814.png

InfProd_815.png

InfProd_816.png

InfProd_817.png

InfProd_818.png

The following approximations hold to about 2% over all a :

InfProd_819.png

Sin[π a] = Cos[π (a - 1/2)] :

InfProd_820.png

InfProd_821.png

InfProd_822.png

InfProd_823.png

InfProd_824.png

InfProd_825.png

InfProd_826.png

InfProd_827.png

InfProd_828.png

InfProd_829.png

InfProd_830.png

InfProd_831.png

InfProd_832.png

InfProd_833.png

InfProd_834.gif

InfProd_835.png

InfProd_836.png

InfProd_837.png

InfProd_838.png

InfProd_839.png

InfProd_840.png

InfProd_841.png

InfProd_842.png

InfProd_843.png

InfProd_844.png

InfProd_845.png

InfProd_846.png

InfProd_847.gif

InfProd_848.png

InfProd_849.png

InfProd_850.png

InfProd_851.png

InfProd_852.png

InfProd_853.png

InfProd_854.png

InfProd_855.png

InfProd_856.png

InfProd_857.png

InfProd_858.png

InfProd_859.gif

InfProd_860.png

InfProd_861.png

InfProd_862.png

InfProd_863.png

InfProd_864.png

InfProd_865.png

InfProd_866.png

InfProd_867.png

InfProd_868.png

InfProd_869.png

InfProd_870.png

InfProd_871.gif

InfProd_872.gif

InfProd_873.png

InfProd_874.png

Amazing identities connecting trigonometric and lemniscate functions:

InfProd_875.png

InfProd_876.png

InfProd_877.png

Special values of trigonometric functions :

Euler :

InfProd_878.png

Sin[π k/n] , n = 2 to 8 (rows) and k = 1 to n - 1 (columns) :

InfProd_879.png

1
InfProd_880.png InfProd_881.png
InfProd_882.png 1 InfProd_883.png
InfProd_884.png InfProd_885.png InfProd_886.png InfProd_887.png
InfProd_888.png InfProd_889.png 1 InfProd_890.png InfProd_891.png
InfProd_892.png InfProd_893.png InfProd_894.png InfProd_895.png InfProd_896.png InfProd_897.png
InfProd_898.png InfProd_899.png InfProd_900.png 1 InfProd_901.png InfProd_902.png InfProd_903.png

Cos[π k/n], n = 2 to 8 (rows) and k = 1 to n - 1 (columns) :

InfProd_904.png

0
InfProd_905.png InfProd_906.png
InfProd_907.png 0 InfProd_908.png
InfProd_909.png InfProd_910.png InfProd_911.png InfProd_912.png
InfProd_913.png InfProd_914.png 0 InfProd_915.png InfProd_916.png
InfProd_917.png InfProd_918.png InfProd_919.png InfProd_920.png InfProd_921.png InfProd_922.png
InfProd_923.png InfProd_924.png InfProd_925.png 0 InfProd_926.png InfProd_927.png InfProd_928.png

Repeated bisection of the angle (2n+1)π inside of trigonometric functions gives (row k = 1 to 6 , column n =  0 to 5):

InfProd_929.png

1
-1
1
-1
1
-1
InfProd_930.png
InfProd_931.png
InfProd_932.png
InfProd_933.png
InfProd_934.png
InfProd_935.png
InfProd_936.png
InfProd_937.png
InfProd_938.png InfProd_939.png
InfProd_940.png
InfProd_941.png
InfProd_942.png
InfProd_943.png
InfProd_944.png
InfProd_945.png
InfProd_946.png
InfProd_947.png
InfProd_948.png
InfProd_949.png
InfProd_950.png
InfProd_951.png
InfProd_952.png
InfProd_953.png
InfProd_954.png
InfProd_955.png
InfProd_956.png
InfProd_957.png
InfProd_958.png
InfProd_959.png

InfProd_960.png

0
0
0
0
0
0
InfProd_961.png
InfProd_962.png
InfProd_963.png
InfProd_964.png
InfProd_965.png
InfProd_966.png
InfProd_967.png
InfProd_968.png
InfProd_969.png
InfProd_970.png
InfProd_971.png
InfProd_972.png
InfProd_973.png
InfProd_974.png
InfProd_975.png
InfProd_976.png
InfProd_977.png
InfProd_978.png
InfProd_979.png
InfProd_980.png
InfProd_981.png
InfProd_982.png
InfProd_983.png
InfProd_984.png
InfProd_985.png
InfProd_986.png
InfProd_987.png
InfProd_988.png
InfProd_989.png
InfProd_990.png

Repeated angular bisection of any angle φ  :

InfProd_991.png

InfProd_992.png InfProd_993.png InfProd_994.png InfProd_995.png InfProd_996.png InfProd_997.png
InfProd_998.png InfProd_999.png InfProd_1000.png InfProd_1001.png InfProd_1002.png InfProd_1003.png
InfProd_1004.png InfProd_1005.png InfProd_1006.png InfProd_1007.png InfProd_1008.png InfProd_1009.png
InfProd_1010.png InfProd_1011.png InfProd_1012.png InfProd_1013.png InfProd_1014.png InfProd_1015.png

If Cos[φ] allows a radical expression (see below), then the Cos or Sin of the repeated bisection InfProd_1016.png  also have radical forms as shown exemplarily  in the next two tables (row k = 0 to 3 bisections , angles φ = 0 to 5π/48 in steps of π/48 (columns)):

InfProd_1017.png

sin[0]
0
InfProd_1018.png
InfProd_1019.png
InfProd_1020.png
InfProd_1021.png
InfProd_1022.png
InfProd_1023.png
InfProd_1024.png
InfProd_1025.png
InfProd_1026.png
InfProd_1027.png
sin[0]
0
InfProd_1028.png
InfProd_1029.png
InfProd_1030.png
InfProd_1031.png
InfProd_1032.png
InfProd_1033.png
InfProd_1034.png
InfProd_1035.png
InfProd_1036.png
InfProd_1037.png
sin[0]
0
InfProd_1038.png
InfProd_1039.png
InfProd_1040.png
InfProd_1041.png
InfProd_1042.png
InfProd_1043.png
InfProd_1044.png
InfProd_1045.png
InfProd_1046.png
InfProd_1047.png
sin[0]
0
InfProd_1048.png
InfProd_1049.png
InfProd_1050.png
InfProd_1051.png
InfProd_1052.png
InfProd_1053.png
InfProd_1054.png
InfProd_1055.png
InfProd_1056.png
InfProd_1057.png

InfProd_1058.png

cos[0]
1
InfProd_1059.png
InfProd_1060.png
InfProd_1061.png
InfProd_1062.png
InfProd_1063.png
InfProd_1064.png
InfProd_1065.png
InfProd_1066.png
InfProd_1067.png
InfProd_1068.png
cos[0]
1
InfProd_1069.png
InfProd_1070.png
InfProd_1071.png
InfProd_1072.png
InfProd_1073.png
InfProd_1074.png
InfProd_1075.png
InfProd_1076.png
InfProd_1077.png
InfProd_1078.png
cos[0]
1
InfProd_1079.png
InfProd_1080.png
InfProd_1081.png
InfProd_1082.png
InfProd_1083.png
InfProd_1084.png
InfProd_1085.png
InfProd_1086.png
InfProd_1087.png
InfProd_1088.png
cos[0]
1
InfProd_1089.png
InfProd_1090.png
InfProd_1091.png
InfProd_1092.png
InfProd_1093.png
InfProd_1094.png
InfProd_1095.png
InfProd_1096.png
InfProd_1097.png
InfProd_1098.png

Radicals for Cos[φ] up to π/2 are for example (in steps of π/24) :

InfProd_1099.png

InfProd_1100.png

or in steps of π/10 :

InfProd_1101.png

InfProd_1102.png

Note that 2Cos[ π/5] and 2Cos[2π/5] are equal to Φ (the golden ratio) and to its inverse.

Tan[π k/n], n = 2 to 8 (rows) and k = 1 to n - 1 (columns) :

InfProd_1103.png

ComplexInfinity
InfProd_1104.png InfProd_1105.png
1 ComplexInfinity -1
InfProd_1106.png InfProd_1107.png InfProd_1108.png InfProd_1109.png
InfProd_1110.png InfProd_1111.png ComplexInfinity InfProd_1112.png InfProd_1113.png
InfProd_1114.png InfProd_1115.png InfProd_1116.png InfProd_1117.png InfProd_1118.png InfProd_1119.png
InfProd_1120.png 1 InfProd_1121.png ComplexInfinity InfProd_1122.png -1 InfProd_1123.png

Cot[π k/n], n = 2 to 8 (rows) and k = 1 to n - 1 (columns) :

InfProd_1124.png

0
InfProd_1125.png InfProd_1126.png
1 0 -1
InfProd_1127.png InfProd_1128.png InfProd_1129.png InfProd_1130.png
InfProd_1131.png InfProd_1132.png 0 InfProd_1133.png InfProd_1134.png
InfProd_1135.png InfProd_1136.png InfProd_1137.png InfProd_1138.png InfProd_1139.png InfProd_1140.png
InfProd_1141.png 1 InfProd_1142.png 0 InfProd_1143.png -1 InfProd_1144.png

Some (special) special values :

InfProd_1145.png InfProd_1146.png
InfProd_1147.png InfProd_1148.png
InfProd_1149.png InfProd_1150.png

InfProd_1151.png

InfProd_1152.png InfProd_1153.png
InfProd_1154.png InfProd_1155.png
InfProd_1156.png InfProd_1157.png

InfProd_1158.png

q - Series :   ( Back to Top )

with InfProd_1159.png → Cosh[ k Log[ q ]] + Sinh[ k Log[ q ]] the following expressions can be transformed into sums of hyperbolic functions.

InfProd_1160.png

The inner sum above gives the number of ascending sequences of length k in the permutations of n numbers.
For natural n PolyLog[-n, q] appears as a rational function in q, the coefficients of q appear as the Eulerian Numbers.

InfProd_1161.png

InfProd_1162.png

InfProd_1163.png

The appearing of n or n - 1 as summation stop index implies n ∈ N.

InfProd_1164.png

There is a small stumble stone in the definition of LerchPhi in the neighbourhood of a = 0: LerchPhi[q, n, a] = InfProd_1165.png, it changes for a = 0 abruptly to a different function InfProd_1166.png.

InfProd_1167.png

InfProd_1168.png

InfProd_1169.png

InfProd_1170.png

InfProd_1171.png

InfProd_1172.png

InfProd_1173.png

InfProd_1174.png

InfProd_1175.png

InfProd_1176.png

InfProd_1177.gif

InfProd_1178.gif

InfProd_1179.png

InfProd_1180.png

InfProd_1181.png

InfProd_1182.png

InfProd_1183.gif

InfProd_1184.png

InfProd_1185.png

InfProd_1186.png

Using an identity from (R4) series involving the Floor function can be solved :

InfProd_1187.png

InfProd_1188.png

InfProd_1189.png

InfProd_1190.png

( m = InverseEllipticNomeQ[q], K[m] = EllipticK[m], E[m] = EllipticE[m] ):

InfProd_1191.png

InfProd_1192.png

InfProd_1193.png

InfProd_1194.png

InfProd_1195.png

InfProd_1196.png

InfProd_1197.png

InfProd_1198.png

InfProd_1199.png

InfProd_1200.png

InfProd_1201.png

Lambert Type q Series:

InfProd_1202.png

InfProd_1203.png

InfProd_1204.png

InfProd_1205.gif

InfProd_1206.png

InfProd_1207.gif

InfProd_1208.gif

InfProd_1209.png

InfProd_1210.png

InfProd_1211.gif

InfProd_1212.png

InfProd_1213.gif

InfProd_1214.gif

The introduction of QPolyGamma[n, z, q] (nth derivative of the QDigamma function (z, q)) in Mathematica 7 allows expression of

InfProd_1215.png

InfProd_1216.png

InfProd_1217.png

InfProd_1218.png

InfProd_1219.png

InfProd_1220.png

InfProd_1221.png

InfProd_1222.png

InfProd_1223.png

InfProd_1224.png

InfProd_1225.png

InfProd_1226.png

InfProd_1227.png

InfProd_1228.png

InfProd_1229.png

InfProd_1230.png

InfProd_1231.png

InfProd_1232.png

InfProd_1233.png

InfProd_1234.png

InfProd_1235.png

InfProd_1236.png

InfProd_1237.png

InfProd_1238.png

InfProd_1239.png

InfProd_1240.png

InfProd_1241.png

InfProd_1242.png

InfProd_1243.png

InfProd_1244.png

InfProd_1245.png

InfProd_1246.png

InfProd_1247.png

InfProd_1248.png

InfProd_1249.png

InfProd_1250.png

InfProd_1251.png

InfProd_1252.png

The next q - series (q → InfProd_1253.png) are connected to the Eisenstein SeriesInfProd_1254.png like

InfProd_1255.png

InfProd_1256.png

InfProd_1257.png

InfProd_1258.png

InfProd_1259.png

InfProd_1260.png

InfProd_1261.png

InfProd_1262.png

With q = InfProd_1263.png this kind of sum is

InfProd_1264.png

InfProd_1265.png

InfProd_1266.png

InfProd_1267.png

InfProd_1268.png

InfProd_1269.png

InfProd_1270.png

InfProd_1271.png

InfProd_1272.png

InfProd_1273.png

InfProd_1274.png

InfProd_1275.png

InfProd_1276.png

InfProd_1277.png

InfProd_1278.png

InfProd_1279.png

InfProd_1280.png

InfProd_1281.png

InfProd_1282.png

InfProd_1283.png

InfProd_1284.png

InfProd_1285.png

InfProd_1286.png

InfProd_1287.png

InfProd_1288.png

InfProd_1289.png

InfProd_1290.png

InfProd_1291.png

InfProd_1292.png

InfProd_1293.png

InfProd_1294.png

InfProd_1295.png

InfProd_1296.png

InfProd_1297.png

InfProd_1298.png

InfProd_1299.png

InfProd_1300.png

InfProd_1301.png

InfProd_1302.png

InfProd_1303.png

InfProd_1304.png

InfProd_1305.png

InfProd_1306.png

For 0.2 < q is in good approximation :

InfProd_1307.png

InfProd_1308.png

InfProd_1309.png

InfProd_1310.png

InfProd_1311.png

InfProd_1312.png

InfProd_1313.png

InfProd_1314.png

InfProd_1315.png

InfProd_1316.png

InfProd_1317.png

InfProd_1318.png

InfProd_1319.png

InfProd_1320.png

InfProd_1321.png

InfProd_1322.png

InfProd_1323.png

InfProd_1324.png

InfProd_1325.png

InfProd_1326.png

InfProd_1327.png

InfProd_1328.png

InfProd_1329.png

InfProd_1330.png

InfProd_1331.png

InfProd_1332.png

InfProd_1333.png

InfProd_1334.png

InfProd_1335.png

InfProd_1336.png

InfProd_1337.png

InfProd_1338.png

InfProd_1339.png

InfProd_1340.png

InfProd_1341.png

InfProd_1342.png

InfProd_1343.png

InfProd_1344.png

InfProd_1345.png

InfProd_1346.png

InfProd_1347.png

InfProd_1348.png

InfProd_1349.png

InfProd_1350.png

InfProd_1351.png

InfProd_1352.png

InfProd_1353.png

InfProd_1354.png

InfProd_1355.png

InfProd_1356.png

InfProd_1357.png

InfProd_1358.png

InfProd_1359.png

InfProd_1360.png

InfProd_1361.png

InfProd_1362.png

InfProd_1363.png

InfProd_1364.png

InfProd_1365.png

InfProd_1366.png

InfProd_1367.png

InfProd_1368.png

InfProd_1369.png

InfProd_1370.gif

InfProd_1371.png

InfProd_1372.png

InfProd_1373.png

InfProd_1374.png

InfProd_1375.png

InfProd_1376.png

InfProd_1377.png

InfProd_1378.png

InfProd_1379.png

InfProd_1380.png

InfProd_1381.png

InfProd_1382.png

InfProd_1383.png

InfProd_1384.png

InfProd_1385.png

InfProd_1386.png

InfProd_1387.png

InfProd_1388.png

InfProd_1389.png

InfProd_1390.png

InfProd_1391.png

InfProd_1392.png

InfProd_1393.png

InfProd_1394.png

InfProd_1395.png

InfProd_1396.png

InfProd_1397.png

InfProd_1398.png

InfProd_1399.png

InfProd_1400.png

InfProd_1401.png

InfProd_1402.png

InfProd_1403.png

InfProd_1404.png

InfProd_1405.png

InfProd_1406.png

InfProd_1407.png

InfProd_1408.png

InfProd_1409.png

InfProd_1410.png

InfProd_1411.png

InfProd_1412.png

InfProd_1413.png

InfProd_1414.png

InfProd_1415.png

InfProd_1416.png

InfProd_1417.png

InfProd_1418.png

InfProd_1419.png

InfProd_1420.png

InfProd_1421.png

InfProd_1422.png

InfProd_1423.png

InfProd_1424.png

InfProd_1425.png

InfProd_1426.png

InfProd_1427.png

InfProd_1428.png

InfProd_1429.png

InfProd_1430.gif

InfProd_1431.png

InfProd_1432.png

InfProd_1433.png

InfProd_1434.gif

InfProd_1435.png

InfProd_1436.png

InfProd_1437.png

InfProd_1438.png

InfProd_1439.png

InfProd_1440.png

InfProd_1441.png

InfProd_1442.png

InfProd_1443.png

InfProd_1444.png

InfProd_1445.png

InfProd_1446.gif

InfProd_1447.png

InfProd_1448.png

InfProd_1449.png

InfProd_1450.png

InfProd_1451.png

InfProd_1452.png

InfProd_1453.png

InfProd_1454.png

InfProd_1455.png

InfProd_1456.png

InfProd_1457.png

InfProd_1458.png

InfProd_1459.png

InfProd_1460.png

InfProd_1461.png

InfProd_1462.png

InfProd_1463.png

InfProd_1464.png

InfProd_1465.png

InfProd_1466.png

InfProd_1467.png

InfProd_1468.png

other :

InfProd_1469.png

InfProd_1470.png

InfProd_1471.png

InfProd_1472.png

InfProd_1473.png

InfProd_1474.png

InfProd_1475.png

InfProd_1476.png

InfProd_1477.png

InfProd_1478.png

InfProd_1479.png

InfProd_1480.png

InfProd_1481.png

InfProd_1482.png

InfProd_1483.png

InfProd_1484.png

InfProd_1485.png

InfProd_1486.png

InfProd_1487.png

InfProd_1488.png

InfProd_1489.png

QFunction Identities :

InfProd_1490.png

InfProd_1491.png

InfProd_1492.png

InfProd_1493.png

InfProd_1494.png

InfProd_1495.png

InfProd_1496.png

Special values of QPolyGamma :

InfProd_1497.png

InfProd_1498.png

InfProd_1499.png

InfProd_1500.gif

InfProd_1501.png

InfProd_1502.png

InfProd_1503.png

InfProd_1504.png

InfProd_1505.png

InfProd_1506.png

InfProd_1507.png

InfProd_1508.png

InfProd_1509.png

InfProd_1510.png

InfProd_1511.png

InfProd_1512.gif

InfProd_1513.png

InfProd_1514.png

InfProd_1515.png

InfProd_1516.png

InfProd_1517.png

InfProd_1518.png

InfProd_1519.png

InfProd_1520.png

InfProd_1521.png

InfProd_1522.png

InfProd_1523.png

InfProd_1524.png

InfProd_1525.png

InfProd_1526.png

InfProd_1527.png

InfProd_1528.png

InfProd_1529.png

InfProd_1530.png

InfProd_1531.png

InfProd_1532.png

InfProd_1533.png

InfProd_1534.png

InfProd_1535.png

InfProd_1536.png

InfProd_1537.png

InfProd_1538.png

InfProd_1539.png

InfProd_1540.png

InfProd_1541.png

InfProd_1542.png

InfProd_1543.png

InfProd_1544.png

InfProd_1545.png

InfProd_1546.png

InfProd_1547.png

InfProd_1548.png

InfProd_1549.png

InfProd_1550.png

InfProd_1551.png

InfProd_1552.png

InfProd_1553.png

InfProd_1554.png

InfProd_1555.png

InfProd_1556.gif

InfProd_1557.png

With x ∈ Reals is   InfProd_1558.png

InfProd_1559.png
Real Part Imaginary Part
n=1: InfProd_1560.png 0
n=2: InfProd_1561.png InfProd_1562.png
n=3: InfProd_1563.png InfProd_1564.png
n=4: InfProd_1565.png InfProd_1566.png
n=5: InfProd_1567.png InfProd_1568.png
n=6: InfProd_1569.png InfProd_1570.png
n=7: InfProd_1571.png InfProd_1572.png

InfProd_1573.png

special values of EllipticK and EllipticE:   ( Back to Top )

K[m] is EllipticK[m];

InfProd_1574.png

E[m] is EllipticE[m];

InfProd_1575.png

InfProd_1576.png

InfProd_1577.png

InfProd_1578.png

InfProd_1579.png

InfProd_1580.png

InfProd_1581.png

InfProd_1582.png

InfProd_1583.png

InfProd_1584.png

InfProd_1585.png

InfProd_1586.png

InfProd_1587.png

InfProd_1588.png

Series of Hyperbolic Functions:   ( Back to Top )

InfProd_1589.png

InfProd_1590.png

InfProd_1591.png

InfProd_1592.png

InfProd_1593.gif

InfProd_1594.gif

InfProd_1595.png

InfProd_1596.png

InfProd_1597.png

InfProd_1598.png

ϑ ‘’ represents the second derivative of ϑ (x,q) with respect to x, eg: InfProd_1599.png means InfProd_1600.png.

InfProd_1601.png

InfProd_1602.png

InfProd_1603.png

InfProd_1604.png

InfProd_1605.png

InfProd_1606.png

The following series containing λ converge very fast with increasing λ :

InfProd_1607.png

InfProd_1608.png

InfProd_1609.png

InfProd_1610.png

InfProd_1611.png

InfProd_1612.png

InfProd_1613.png

InfProd_1614.png

InfProd_1615.png

InfProd_1616.png

InfProd_1617.png

InfProd_1618.png

InfProd_1619.png

InfProd_1620.png

InfProd_1621.png

InfProd_1622.png

InfProd_1623.png

InfProd_1624.png

InfProd_1625.png

InfProd_1626.png

InfProd_1627.png

InfProd_1628.png

InfProd_1629.png

InfProd_1630.png

InfProd_1631.png

InfProd_1632.png

InfProd_1633.png

InfProd_1634.png

InfProd_1635.png

InfProd_1636.png

InfProd_1637.png

InfProd_1638.png

Some Jacobi elliptic functions :

InfProd_1639.png

InfProd_1640.png

InfProd_1641.png

InfProd_1642.gif

InfProd_1643.png

Hyperbolic series involving the lemniscate functions :

ϖ is the lemniscate constant :

InfProd_1644.png

InfProd_1645.png

Connected to the above series are fast converging expansions of special elliptic functions that equal sl[x] and cl[x], the lemniscate sine and cosine, useful for numeric computation of sl and cl:

InfProd_1646.gif

The first 5 Taylor coefficients of the sum representing sl for increasing index m, see sequence A104203 in OEIS (1, -12, 3024, -4390848, 21224560896,...) :

InfProd_1647.png

0 1.0494342235. -11.900183700. 3023.8648359. -4.3909020798.*^6 2.1224555712.*^10
3 0.99999600967. -12.000008223. 3023.9999831. -4.3908480000.*^6 2.1224560896.*^10
6 1.0000000003. -11.999999999. 3024.0000000. -4.3908480000.*^6 2.1224560896.*^10
9 1.000000000. -12.000000000. 3024.0000000. -4.3908480000.*^6 2.1224560896.*^10

The first 8 Taylor coefficients of the sum representing cl for increasing index m, compare sequence A159600 in OEIS (1, -1, 3, -27, 441, -11529, 442827, -23444883,... but watch out,  A159600 excludes a factorInfProd_1648.png needed to obtain the series for cl) :

InfProd_1649.png

0 1.0412730250. -1.0296444931. 3.0214068820. -27.016199247. 441.01703299. -11529.047467. 442827.26751. -2.3444884804.*^7
3 0.99999666957. -0.99999760952. 2.9999982842. -26.999998768. 440.99999912. -11528.999999. 442827.00000. -2.3444883000.*^7
6 1.0000000003. -1.0000000002. 3.0000000001. -27.000000000. 441.00000000. -11529.000000. 442827.00000. -2.3444883000.*^7
9 1.000000000. -1.000000000. 3.0000000000. -27.000000000. 441.00000000. -11529.000000. 442827.00000. -2.3444883000.*^7

Near the real axis the lemniscate functions may be described by Fourier series :

InfProd_1650.gif

InfProd_1651.png

Ramanujan's Cos/Cosh identity :

InfProd_1652.png

The approximation of the the next two series to the lemniscate functions in the area  around the origin improves with the number of included terms in the numerical evaluation (more terms or ‘building blocks‘ cover a larger domain):

InfProd_1653.png

InfProd_1654.png

InfProd_1655.png

The 'four horsemen of the apocalypse':

InfProd_1656.png

InfProd_1657.png

W A I T !  Have you noticed the beauty of  the series above? The alternating sum of shifted 1/Cosh functions gives the lemniscate cosine, the alternating sum of shifted 1/Sinh functions gives 1/(lemniscate sine)... Isn’t that incredible?

InfProd_1658.png

InfProd_1659.png

InfProd_1660.png

InfProd_1661.png

InfProd_1662.png

InfProd_1663.png

InfProd_1664.png

InfProd_1665.png

InfProd_1666.png

InfProd_1667.png

InfProd_1668.png

InfProd_1669.png

InfProd_1670.png

InfProd_1671.png

The next two series results show a sign flip at every other integer interval in y, shifted by 1/2, due to the complex square root. This switch of sign is taken into account by the factor (-1)^Floor[ y - 1/2 ].

InfProd_1672.png

InfProd_1673.png

InfProd_1674.png

InfProd_1675.png

InfProd_1676.png

InfProd_1677.png

InfProd_1678.png

The QPolyGamma 'monsters' :

InfProd_1679.png

InfProd_1680.png

InfProd_1681.png

InfProd_1682.png

InfProd_1683.png

InfProd_1684.png

InfProd_1685.png

InfProd_1686.png

InfProd_1687.png

InfProd_1688.png

InfProd_1689.png

m = InverseEllipticNomeQ[InfProd_1690.png] :

InfProd_1691.png

InfProd_1692.png

InfProd_1693.png

InfProd_1694.png

InfProd_1695.png

InfProd_1696.png

InfProd_1697.png

InfProd_1698.png

InfProd_1699.png

InfProd_1700.png

InfProd_1701.png

InfProd_1702.png

InfProd_1703.png

InfProd_1704.png

InfProd_1705.png

InfProd_1706.png

InfProd_1707.png

InfProd_1708.png

InfProd_1709.png

InfProd_1710.png

InfProd_1711.png

InfProd_1712.png

InfProd_1713.png

InfProd_1714.png

The next expression uses an idea taken from “Weiss, J.D.(2014) The Summation of One Class of Infinite Series. Applied Mathematics, 5, 2815 - 2822. http://dx.doi.org/10.4236/am.2014.517269” :

InfProd_1715.png

both series approach - Log[2] from either below (Coth) or above (Tanh) for increasing z .

InfProd_1716.png

InfProd_1717.png

InfProd_1718.png

InfProd_1719.png

InfProd_1720.png

InfProd_1721.png

InfProd_1722.png

InfProd_1723.png

InfProd_1724.png

both series above approach z / (1 - z) from either below (Tanh) or above (Coth) for 1 < x.

InfProd_1725.png

InfProd_1726.png

both series above approach PolyLog[-j,z] from either below (Tanh) or above (Coth) for 1 < x.

InfProd_1727.png

InfProd_1728.png

InfProd_1729.png

InfProd_1730.png

m = InverseEllipticNomeQ[InfProd_1731.png] :

InfProd_1732.png

InfProd_1733.png

The real parts of the next four series are well defined for (-1 < x). The real parts for 0 < x and the imaginary Parts for all x converge very fast for small m.

InfProd_1734.png

InfProd_1735.png

InfProd_1736.png

InfProd_1737.png

InfProd_1738.png

InfProd_1739.png

InfProd_1740.png

InfProd_1741.png

InfProd_1742.png

InfProd_1743.png

InfProd_1744.png

InfProd_1745.png

InfProd_1746.png

InfProd_1747.png

InfProd_1748.png

InfProd_1749.png

InfProd_1750.png

InfProd_1751.png

InfProd_1752.png

InfProd_1753.png

InfProd_1754.png

InfProd_1755.png

InfProd_1756.png

InfProd_1757.png

InfProd_1758.png

InfProd_1759.png

InfProd_1760.png

InfProd_1761.png

InfProd_1762.png

InfProd_1763.png

InfProd_1764.png

InfProd_1765.png

InfProd_1766.png

InfProd_1767.png

InfProd_1768.png

InfProd_1769.png

m = InverseEllipticNomeQ[InfProd_1770.png] :

InfProd_1771.png

InfProd_1772.png

InfProd_1773.png

InfProd_1774.png

InfProd_1775.png

InfProd_1776.png

Some hyperbolic Identities :

InfProd_1777.png

InfProd_1778.png

InfProd_1779.gif

InfProd_1780.gif

Some Lemniscate Sine and Cosine Identities including derivative and integral:

Periods :

InfProd_1781.png

Dual sibling of the Pythagorean Identity InfProd_1782.png) :

InfProd_1783.png

Special values :

InfProd_1784.png

InfProd_1785.png

InfProd_1786.png

InfProd_1787.png

InfProd_1788.png

Argument addition formulae :

InfProd_1789.png

Imaginary, negative and double arguments :

InfProd_1790.png

Squares :

InfProd_1791.png

Derivatives and basic integrals :

InfProd_1792.gif

From the argument addition formulae for the lemniscate functions taking  dx small with sl[dx] = dx and cl[dx] = 1 we get the derivative as sl'[x] =InfProd_1793.png (sl[x + dx] - sl[x])/dx = cl[x] (1 + InfProd_1794.png) and likewise for cl'[x].
The integrals may then so be checked by calculating the derivatives of their right hand sides. The expressions show some similarity (duality) with their trigonometric counterparts.

For more integrals of the lemniscate functions scroll to the bottom of :

Table of Indefinite Integrals

The following identities can be verified using the imaginary and symmetry properties of sl and cl given above.

InfProd_1797.png

Complex properties :

InfProd_1798.png

Product approximations to the lemniscate functions (for 0 < m both better than InfProd_1799.png for all x, they  converge more than twice as fast as the already fast converging series given above in the section) :

InfProd_1800.png

The ‘lemniscatic tangent’ is then represented by :

InfProd_1801.png

Lemniscate functions as special cases  InfProd_1802.png of Jacobi elliptic functions  or Neville theta functions:

InfProd_1803.png

InfProd_1804.png

Series of CosIntegral:   ( Back to Top )

InfProd_1805.png

InfProd_1806.png

InfProd_1807.png

InfProd_1808.png

InfProd_1809.png

InfProd_1810.png

InfProd_1811.png

InfProd_1812.png

InfProd_1813.png

InfProd_1814.png

InfProd_1815.png

InfProd_1816.png

InfProd_1817.png

InfProd_1818.png

Sums involving reciprocal multifactorials or factorials:   ( Back to Top )

InfProd_1819.png

InfProd_1820.png

InfProd_1821.png

InfProd_1822.png

InfProd_1823.png

InfProd_1824.png

InfProd_1825.png

For even m = 2j  the right hand side reduces to

InfProd_1826.png

InfProd_1827.png

Higher multifactorials :

In the next equations j designates the number of ! within the multifactorial InfProd_1828.png, γ[a, b] is the lower incomplete Gamma Function Γ[a, 0, b] = InfProd_1829.png .

InfProd_1830.png

InfProd_1831.png

InfProd_1832.png

InfProd_1833.png

InfProd_1834.png

Expressions for still higher orders of k may be obtained by applying InfProd_1835.png to both sides of the equation.

The general expression at x = ±1 is, using recursionally defined coefficients  InfProd_1836.png and InfProd_1837.png (the factor after InfProd_1838.png shows the series result for n = 0) :

InfProd_1839.png

where the recursions

InfProd_1840.png

and

InfProd_1841.png

give the integers InfProd_1842.png and InfProd_1843.png, needed for the calculation of the sum above .

The coefficients obtained with low indices j, n (j counting rows from 1 to 8, n counting columns from 0 to 9) are shown here for the regular sum (+1) and the alternating sum (-1):

InfProd_1844.png

InfProd_1845.png

The numbers InfProd_1846.png and InfProd_1847.png represent the Bell numbers BellB[n, 1] and the complementary Bell numbers BellB[n, -1].

InfProd_1848.png

InfProd_1849.png

InfProd_1850.png

Bell' s polynomes are connected to Stirling numbers of the second kind :

InfProd_1851.png

or  may be calculated as higher derivatives of the exponential function : BellB[n, z] InfProd_1852.png  :

InfProd_1853.png

1 1 1 1
z z z -z
InfProd_1854.png InfProd_1855.png InfProd_1856.png InfProd_1857.png
InfProd_1858.png InfProd_1859.png InfProd_1860.png InfProd_1861.png
InfProd_1862.png InfProd_1863.png InfProd_1864.png InfProd_1865.png
InfProd_1866.png InfProd_1867.png InfProd_1868.png InfProd_1869.png
InfProd_1870.png InfProd_1871.png InfProd_1872.png InfProd_1873.png

InfProd_1874.png

InfProd_1875.png

InfProd_1876.png

A completely crazy series :

InfProd_1877.png

The expression in large brackets represents a polynomial in y and InfProd_1878.png of degree n with integer coefficients.

The sum over s above containing Stirling Numbers of the second kind has this structure:

InfProd_1879.png

and appears in a special combinatorics problem :
It calculates the count of possible ways InfProd_1880.png to distribute a number of a differently colored balls into b indistinguishable boxes, so that each box contains two balls or more (see A008299 in OEIS).
The triangular table below lists, how many configurations exist for a = 4 … 20 balls (row#) to go into b = 2 … a/2 boxes (column#), min. 2 per box,
also known as associated Stirling numbers of the second kind :

Table[BB[a, b], {a, 4, 14}, {b, 2, Floor[a/2]}] // TableForm

3
10
25 15
56 105
119 490 105
246 1918 1260
501 6825 9450 945
1012 22935 56980 17325
2035 74316 302995 190575 10395
4082 235092 1487200 1636635 270270
8177 731731 6914908 12122110 4099095 135135

As an aside:
These numbers can also be computed from sums containing products of binomial coefficients (‘n choose k’), divided possibly by factorials as explained in the following example :

Consider 8 differently colored balls to go into 3 boxes. First find IntegerPartitions  of 8 into 3 integers to see the possible partitions of the balls into the boxes :

IntegerPartitions[8, {3}]

InfProd_1881.png

Select the partitions with every element larger than one → {4, 2, 2} and {3, 3, 2} . Start by putting 4 balls into any emtpy box (8 choose 4, Binomial[8, 4] possibilities)  AND choose 2 balls for another empty box (Binomial[8-4, 2])  AND again 2 balls for the last still empty box (Binomial[8-4-2, 2]). Multiply (AND condition) the binomials. Since two elements are equal (2, 2), divide this term by 2!.
Now add  (OR condition) the term from the next partition: put 3 balls into one emtpy box (Binomial[8, 3])  AND 3 balls in another empty box (Binomial[8-3, 3])  AND again 2 balls into the last empty box (Binomial[8-3-3, 2]) and multiply. Divide also this term by the factorial of the number of equal elements. The first argument of the binomials shows the number of ‘unboxed’ balls still to choose from, the second argument contains the element of the partition. All binomials (each corresponds to a box) of a partition are multiplied and (as the possible arrangements of  boxes with an equal number of balls inside are not distinguished)  divided by the factorial of their multiplicity :

InfProd_1882.png

This result corresponds to triangle entry (a = 8, b = 3) .

The ‘crazy’ series above has been looked at in more detail in : Vigren E .; Dieckmann A .; A New Result in Form of Finite Triple Sums for a Series from Ramanujan' s Notebooks . Symmetry 2022, 14, 1090.

InfProd_1883.png

InfProd_1884.png

Higher powers of the factorial in the denominator :

InfProd_1885.gif

Replace InfProd_1886.png with InfProd_1887.png and InfProd_1888.png  with InfProd_1889.png
as well as  InfProd_1890.png with InfProd_1891.png if all symbolic expressions are to be kept strictly real .
The numerical evaluation  of the results above should work over the whole x - range anyway , cancelling possibly imaginary contributions.

Expressions for still higher orders of k above may be obtained by applying (InfProd_1892.png) to both sides of the equation .

The solution of the general series seems more complicated with a single, but quite messy hypergeometric function, (Table[n,k] with k ≤ 0 gives the empty set {}):

InfProd_1893.png

There are cases, where this bulky result may be reduced to simpler functions :
For 2 < m   and for low j the series is given by Mathematica as a somewhat shorter hypergeometric function than in the general case above :

0 = j :

InfProd_1894.png

0 < j < m:

InfProd_1895.png

m ≤ j :
These can be done at the point x = ±1, where a recursion may be calculated, via the next relation that decomposes a reciprocal multifactorial sum containing InfProd_1896.pnginto sums of powers InfProd_1897.png less than InfProd_1898.png:

InfProd_1899.png

and the recursion for the coefficients c is given by :

InfProd_1900.png

obtained with the Mathematica code in the next line …(adapt the signs for the alternating case):

InfProd_1901.png

For example with n = 6, m = 3 :

InfProd_1902.png

InfProd_1903.png

The InfProd_1904.pngs are then :

InfProd_1905.png

1 0 0 1 1 1 2 6 17
0 1 0 0 1 2 3 5 12
0 0 1 0 0 1 3 6 11

Now with s = {0, 1, 2} there is

InfProd_1906.png

InfProd_1907.png

so that the recursionally defined sum yields :

InfProd_1908.png

while Mathematica gives:

InfProd_1909.png

In this way one may derive from the two expressions above reductions of a more complicated hypergeometric function into a series of simpler ones like:

HypergeometricPFQ[{2, 2, 2}, {1, 1, 1, 1, 1}, 1] = 2  HypergeometricPFQ[{}, {1, 1}, 1] + 3 HypergeometricPFQ[{}, {2, 2}, 1] + 3 HypergeometricPFQ[{}, {1, 2}, 1];

The following  identity (valid for all {m, n ∈ N}) expresses the sum without recursion in terms of sums with exponents of k that are reduced from n to ≤ n-m :

InfProd_1910.png

The  highest exponent of InfProd_1911.png appearing in the recursion above is: if (n < m)  then {s = n} else {s = Min[n-m, m-1]}.

For m = 2 the situation is easier, because all sums (0 ≤ n)InfProd_1912.png can be expressed with Bessel functions through the recursion (see above, set again x = 1)
as linear combinations of the one or two lowest results in the regular case with { {j = 0, BesselI[0, 2]}, {j = 1, BesselI[1, 2]} } up to n = 15:

InfProd_1913.png

1 0 1 1 2 5 13 36 109 359 1266 4731 18657 77464 337681 1540381
0 1 0 1 2 4 10 29 90 295 1030 3838 15168 63117 275252 1254801
n InfProd_1914.png
0 BesselI[0,2]
1 BesselI[1,2]
2 BesselI[0,2]
3 BesselI[0,2]+BesselI[1,2]
4 2 BesselI[0,2]+2 BesselI[1,2]
5 5 BesselI[0,2]+4 BesselI[1,2]
6 13 BesselI[0,2]+10 BesselI[1,2]
7 36 BesselI[0,2]+29 BesselI[1,2]
8 109 BesselI[0,2]+90 BesselI[1,2]
9 359 BesselI[0,2]+295 BesselI[1,2]
10 1266 BesselI[0,2]+1030 BesselI[1,2]
11 4731 BesselI[0,2]+3838 BesselI[1,2]
12 18657 BesselI[0,2]+15168 BesselI[1,2]
13 77464 BesselI[0,2]+63117 BesselI[1,2]
14 337681 BesselI[0,2]+275252 BesselI[1,2]
15 1540381 BesselI[0,2]+1254801 BesselI[1,2]

or in the alternating case with { {j = 0, BesselJ[0, 2]}, {j = 1, - BesselJ[1, 2]} }:

InfProd_1915.png

1 0 -1 -1 0 3 9 16 7 -87 -472 -1567 -3375 -216 45927 308107
0 1 0 -1 -2 -2 2 17 54 109 54 -796 -5000 -19499 -52252 -44617
n InfProd_1916.png
0 BesselJ[0,2]
1 -BesselJ[1,2]
2 -BesselJ[0,2]
3 -BesselJ[0,2]+BesselJ[1,2]
4 2 BesselJ[1,2]
5 3 BesselJ[0,2]+2 BesselJ[1,2]
6 9 BesselJ[0,2]-2 BesselJ[1,2]
7 16 BesselJ[0,2]-17 BesselJ[1,2]
8 7 BesselJ[0,2]-54 BesselJ[1,2]
9 -87 BesselJ[0,2]-109 BesselJ[1,2]
10 -472 BesselJ[0,2]-54 BesselJ[1,2]
11 -1567 BesselJ[0,2]+796 BesselJ[1,2]
12 -3375 BesselJ[0,2]+5000 BesselJ[1,2]
13 -216 BesselJ[0,2]+19499 BesselJ[1,2]
14 45927 BesselJ[0,2]+52252 BesselJ[1,2]
15 308107 BesselJ[0,2]+44617 BesselJ[1,2]

A generalization to powers of multifactorials :

InfProd_1917.png

where two recursions are needed:

InfProd_1918.gif

InfProd_1919.png

The series

InfProd_1920.png

converge very fast (the terms k > 3 contribute less than InfProd_1921.png) . For m = 3  the terms give :

InfProd_1922.png

With n = 0 the first sum and (InfProd_1923.png- the second sum) are very close to the  Pomerance Number  1.25002143347...(A100085 in Oeis).

Multiple Sums (lattice sums) :   ( Back to Top )

Factorial :

InfProd_1924.png

InfProd_1925.png

InfProd_1926.png

InfProd_1927.png

InfProd_1928.png

InfProd_1929.png

InfProd_1930.png

InfProd_1931.png

InfProd_1932.png

InfProd_1933.png

InfProd_1934.png

InfProd_1935.png

InfProd_1936.png

InfProd_1937.png

InfProd_1938.png

InfProd_1939.png

InfProd_1940.png

InfProd_1941.png

InfProd_1942.png

Borwein' s formula :

InfProd_1943.png

Benson' s formula:

InfProd_1944.png

already the sum for all {m, n} < 5 gives a very good approximation .

InfProd_1945.png

InfProd_1946.png

InfProd_1947.png

InfProd_1948.png

InfProd_1949.png

InfProd_1950.png

InfProd_1951.png

InfProd_1952.png

InfProd_1953.png

InfProd_1954.png

InfProd_1955.png

InfProd_1956.png

InfProd_1957.png

InfProd_1958.png

InfProd_1959.png

InfProd_1960.png

InfProd_1961.png

InfProd_1962.png

InfProd_1963.png

The approximations for the first two sums in the two following lines are for 2 < m better than 10^-10, the Cot Csc term corresponds to the dominant j = 0 term in the double sum.

InfProd_1964.png

InfProd_1965.png

Double series involving lemniscate sine and cosine :

InfProd_1966.png

InfProd_1967.png

InfProd_1968.png

The sums above show only asymmetric convergence, i.e.  the range of summation index i must be much larger than the one of j, the convergence of the imaginary part is faster.

InfProd_1969.png

Some values of Eisenstein series :

InfProd_1970.png

InfProd_1971.gif

The coefficients of InfProd_1972.pngwritten as hn[4s], (s = 2, 3,…) are called Hurwitz numbers and may be calculated by the following recurrence:

InfProd_1973.png

InfProd_1974.png

or directly by :

InfProd_1975.png

InfProd_1976.gif

InfProd_1977.png

For s = 6, 10, 14, … the above right hand side of the equation yields zero.

InfProd_1978.png

InfProd_1979.png

For s = 6, 10, 14, …  at x = 1 the right hand side of the equation above yields zero.

InfProd_1980.png

InfProd_1981.png

InfProd_1982.png

InfProd_1983.png

InfProd_1984.png

InfProd_1985.png

InfProd_1986.png

InfProd_1987.png

InfProd_1988.png

some of the next series appear also in electrostatic problems, see
Vigren E.; Dieckmann A.; Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges. Symmetry 2020, 12, 1040.

InfProd_1989.png

InfProd_1990.png

InfProd_1991.png

InfProd_1992.png

InfProd_1993.png

InfProd_1994.png

InfProd_1995.png

The encounter of lemniscate functions with electrostatics suggests a tribute to C.F.Gauss, who paved the way for a better understanding of so many problems.

InfProd_1996.png

InfProd_1997.png

InfProd_1998.png

The next double Series converges (asymmetric convergence) best numerically if i >> j.

InfProd_1999.png

Double Series involving the Zeta function :

InfProd_2000.png

InfProd_2001.png

InfProd_2002.png

InfProd_2003.gif

InfProd_2004.png

InfProd_2005.png

InfProd_2006.png

Multiple Series involving the Zeta function :

InfProd_2007.gif

InfProd_2008.png

For a few values of s = 2 to 6 (rows) and m = 1 to s - 1 (columns) the expansions of the right hand side above are given :

InfProd_2009.png
Zeta[3] InfProd_2010.png
InfProd_2011.png InfProd_2012.png InfProd_2013.png
Zeta[5] InfProd_2014.png InfProd_2015.png InfProd_2016.png
InfProd_2017.png InfProd_2018.png InfProd_2019.png InfProd_2020.png InfProd_2021.png

InfProd_2022.png

For a few values of s = 2 to 7 (rows) and m = 1 to s - 1 (columns) the expansions of the right hand side above are given (for m = 1 replace the returned 0’s with Zeta[s]) :

InfProd_2023.png
Zeta[3] InfProd_2024.png
InfProd_2025.png Zeta[3] InfProd_2026.png
Zeta[5] InfProd_2027.png InfProd_2028.png InfProd_2029.png
InfProd_2030.png Zeta[5] InfProd_2031.png InfProd_2032.png InfProd_2033.png
Zeta[7] InfProd_2034.png InfProd_2035.png InfProd_2036.png InfProd_2037.png InfProd_2038.png

The notation InfProd_2039.pngmeans that the divergent term 1/0 is excluded :

InfProd_2040.png

For a few values of  s = 1 to 6 (rows) and m = 1 to 5 (columns)  the expansions of the right hand side above are given :

Log[2] InfProd_2041.png InfProd_2042.png InfProd_2043.png InfProd_2044.png
InfProd_2045.png InfProd_2046.png InfProd_2047.png InfProd_2048.png InfProd_2049.png
InfProd_2050.png InfProd_2051.png InfProd_2052.png InfProd_2053.png InfProd_2054.png
InfProd_2055.png InfProd_2056.png InfProd_2057.png InfProd_2058.png InfProd_2059.png
InfProd_2060.png InfProd_2061.png InfProd_2062.png InfProd_2063.png InfProd_2064.png
InfProd_2065.png InfProd_2066.png InfProd_2067.png InfProd_2068.png InfProd_2069.png

InfProd_2070.png

where the number of numerically equal summands  InfProd_2071.png) in the first sum is counted by the multiplicity factor in large brackets within the second sum;

InfProd_2072.png

InfProd_2073.png

s controls the number of ‘slots’ inside the curly brackets of the Hypergeometric function, e.g. :

s = 4; HypergeometricPFQ[Join[Table[1, {k, 1, s}], {m}], Table[2, {k, 1, s}], -1]

InfProd_2074.png

For a few values of  s = 1 to 6 (rows) and m = 1 to 5 (columns) the expansions of the hypergeometric result above are given :

Table[HypergeometricPFQ[Join[Table[1, {k, 1, s}], {m}], Table[2, {k, 1, s}], -1], {s, 1, 6}, {m, 1, 5}] // FullSimplify // PowerExpand // Expand // TableForm

Log[2] InfProd_2075.png InfProd_2076.png InfProd_2077.png InfProd_2078.png
InfProd_2079.png Log[2] InfProd_2080.png InfProd_2081.png InfProd_2082.png
InfProd_2083.png InfProd_2084.png InfProd_2085.png InfProd_2086.png InfProd_2087.png
InfProd_2088.png InfProd_2089.png InfProd_2090.png InfProd_2091.png InfProd_2092.png
InfProd_2093.png InfProd_2094.png InfProd_2095.png InfProd_2096.png InfProd_2097.png
InfProd_2098.png InfProd_2099.png InfProd_2100.png InfProd_2101.png InfProd_2102.png

InfProd_2103.png

InfProd_2104.png

For a few values of s = 1 to 7 (rows) and m = 1 to 5 (columns)  the  results for the series above are given using a summation, that avoids ‘indeterminate’ answers.
Now the multiplicity of numerically equal summands  InfProd_2105.png) is determined by Binomial[k - 1, m - 1]:

InfProd_2106.png

-Log[2] InfProd_2107.png InfProd_2108.png InfProd_2109.png InfProd_2110.png
InfProd_2111.png InfProd_2112.png InfProd_2113.png InfProd_2114.png InfProd_2115.png
InfProd_2116.png InfProd_2117.png InfProd_2118.png InfProd_2119.png InfProd_2120.png
InfProd_2121.png InfProd_2122.png InfProd_2123.png InfProd_2124.png InfProd_2125.png
InfProd_2126.png InfProd_2127.png InfProd_2128.png InfProd_2129.png InfProd_2130.png
InfProd_2131.png InfProd_2132.png InfProd_2133.png InfProd_2134.png InfProd_2135.png
InfProd_2136.png InfProd_2137.png InfProd_2138.png InfProd_2139.png InfProd_2140.png

InfProd_2141.png

For the lowest values of s and m this sum is:

InfProd_2142.png

and

InfProd_2143.png

For the lowest values of s and m this sum is:

InfProd_2144.png

InfProd_2145.png

For the lowest values of s and m this sum is :

InfProd_2146.png

InfProd_2147.png

For the lowest values of s and m this sum is :

InfProd_2148.png

Many of the series found in this table are connected to a ' lattice version' like (try it!) :

InfProd_2149.png

For instance  (find an identity with start index 0, replace  kInfProd_2150.png ,  insert the Gammas, the j - 1 factorial  and sum over all i's) :

InfProd_2151.png

For a series  InfProd_2152.png starting with index 1 it is a little more involved, because the first lattice summand will be InfProd_2153.png :

InfProd_2154.png

So (for example) the lattice version of

InfProd_2155.png

InfProd_2156.png

Lattice q - sums :

InfProd_2157.png

InfProd_2158.png

InfProd_2159.png

Following ideas of (R4) look at the m-dimensional lattice sum

InfProd_2160.png

where the summands characterized by  InfProd_2161.png = k occur with a certain multiplicity InfProd_2162.png given by

InfProd_2163.png

This tells the number of ways to express k as a sum of m integers InfProd_2164.png (how often a certain term ' k' occurs) ,
for instance  (k = 3, m = 2; 4 ways) : 3 = 0 + 3 or 3 + 0 or 1 + 2 or 2 + 1.
Then the lattice sum can be reduced to a single sum like

InfProd_2165.png

which gives after multiplication with (m - 1)! and evaluation

InfProd_2166.png

This identity can be used to successively get values of InfProd_2167.png. It  determines InfProd_2168.png at m = 3:

InfProd_2169.png

Expanding the sum over (k+2) shows InfProd_2170.png and InfProd_2171.png, and if it is assumed that InfProd_2172.png = 1/2 (regularization), then it follows that InfProd_2173.png = - 1/4.

For m = n + 2 the sum reads :

InfProd_2174.png

The product inside the sum may be decomposed into a double series of StirlingS1 numbers :

InfProd_2175.png

2+k 2+k
InfProd_2176.png InfProd_2177.png
InfProd_2178.png InfProd_2179.png
InfProd_2180.png InfProd_2181.png
InfProd_2182.png InfProd_2183.png

Isolating the term in (**) with the highest exponent (set the stop index in the sum over j in the table to n - 1) now allows a recursive calculation of the InfProd_2184.pnglike:

InfProd_2185.png

InfProd_2186.png

Shown above are the InfProd_2187.png for n from 0 to 10 together with the results of corresponding Mathematica sums employing ' Abel' regularization as well as the symbolic HurwitzZeta given at the start of the paragraph.

InfProd_2188.png

diverse Series :   ( Back to Top )

InfProd_2189.png

InfProd_2190.png

InfProd_2191.png

InfProd_2192.png

InfProd_2193.png

InfProd_2194.png

InfProd_2195.png

InfProd_2196.png

InfProd_2197.png

The appearing of n or m as summation stop index implies n, m ∈ N.

InfProd_2198.png

InfProd_2199.png

InfProd_2200.png

InfProd_2201.png

InfProd_2202.png

InfProd_2203.png

InfProd_2204.png

InfProd_2205.png

InfProd_2206.png

InfProd_2207.png

InfProd_2208.png

InfProd_2209.png

InfProd_2210.png

InfProd_2211.png

InfProd_2212.png

InfProd_2213.png

InfProd_2214.png

InfProd_2215.png

InfProd_2216.png

InfProd_2217.png

InfProd_2218.png

InfProd_2219.png

InfProd_2220.png

InfProd_2221.png

InfProd_2222.png

InfProd_2223.png

InfProd_2224.png

InfProd_2225.png

InfProd_2226.png

InfProd_2227.png

InfProd_2228.png

InfProd_2229.png

InfProd_2230.png

InfProd_2231.gif

InfProd_2232.png

InfProd_2233.png

InfProd_2234.png

InfProd_2235.png

InfProd_2236.png

InfProd_2237.png

InfProd_2238.png

Sum of the inverse m - gonal numbers :

InfProd_2239.png

InfProd_2240.png

Values of the series for the first m :

InfProd_2241.png

The sum of  integer powers  of the inverse m - gonal numbers :

InfProd_2242.png

The quadratic case m = 4, where the formula above gives an indeterminate result,  may be calculated as :

InfProd_2243.png

k - nomial triangles:

Generate the  k - nomial triangle  as coefficents of InfProd_2244.png:
(the mth coefficient in the nth row  gives the frequency of the sum of points with value m + n - 2, shown after a throw of n - 1  fair k-sided dice; displayed are the cases k = {2 bi-, 3 tri-, 4 quadrinomial}, up to n = 5)
InfProd_2245.gif

other series:

The sum  InfProd_2246.png  gives following results for some rational s = p/q :

InfProd_2247.png

InfProd_2248.png

InfProd_2249.png

InfProd_2250.png

InfProd_2251.gif

InfProd_2252.png

InfProd_2253.png

InfProd_2254.png

InfProd_2255.png

InfProd_2256.png

InfProd_2257.png

InfProd_2258.png

InfProd_2259.png

InfProd_2260.png

InfProd_2261.png

InfProd_2262.png

InfProd_2263.png

InfProd_2264.png

InfProd_2265.png

InfProd_2266.png

InfProd_2267.png

InfProd_2268.png

InfProd_2269.png

InfProd_2270.png

InfProd_2271.png

InfProd_2272.png

InfProd_2273.png

InfProd_2274.png

InfProd_2275.png

InfProd_2276.png

InfProd_2277.png

InfProd_2278.png

InfProd_2279.png

This sum alternates between ± π  for z ∈ N :

InfProd_2280.png

In the following 4 expressions b =InfProd_2281.png :

InfProd_2282.png

InfProd_2283.png

InfProd_2284.png

InfProd_2285.png

InfProd_2286.png

InfProd_2287.png

InfProd_2288.png

InfProd_2289.png

InfProd_2290.png

InfProd_2291.png

InfProd_2292.png

The next three expressions contain s = InfProd_2293.png and t = InfProd_2294.png:

InfProd_2295.png

InfProd_2296.png

InfProd_2297.png

InfProd_2298.png

InfProd_2299.png

InfProd_2300.png

InfProd_2301.png

InfProd_2302.png

InfProd_2303.png

InfProd_2304.png

InfProd_2305.png

InfProd_2306.png

InfProd_2307.png

InfProd_2308.png

InfProd_2309.png

During a visit in London Leibniz was asked by Huygens to evaluate the sum InfProd_2310.png. He knew about partial fraction decomposition (see next line) and solved it. (Hirsch, der berühmte Herr Leibniz : eine Biographie)

InfProd_2311.png

InfProd_2312.png

InfProd_2313.png

InfProd_2314.png

The sum of  integer powers  of the inverse trigonal numbers :

InfProd_2315.png

InfProd_2316.png

InfProd_2317.png

InfProd_2318.png

InfProd_2319.png

InfProd_2320.png

InfProd_2321.png

InfProd_2322.png

InfProd_2323.png

InfProd_2324.png

InfProd_2325.png

InfProd_2326.png

InfProd_2327.png

InfProd_2328.png

InfProd_2329.png

InfProd_2330.png

InfProd_2331.png

InfProd_2332.png

InfProd_2333.png

InfProd_2334.png

InfProd_2335.png

InfProd_2336.png

InfProd_2337.png

InfProd_2338.png

InfProd_2339.png

InfProd_2340.png

InfProd_2341.png

InfProd_2342.png

InfProd_2343.png

InfProd_2344.png

InfProd_2345.png

InfProd_2346.png

InfProd_2347.png

InfProd_2348.png

InfProd_2349.png

InfProd_2350.png

InfProd_2351.png

InfProd_2352.png

Series of Logarithms :   ( Back to Top )

(m = InverseEllipticNomeQ[q], K[m] = EllipticK[m], E[m] = EllipticE[m]), the appearing of n or n - 1 as summation stop index implies n ∈ N.

InfProd_2353.png

InfProd_2354.png

InfProd_2355.png

InfProd_2356.png

InfProd_2357.png

InfProd_2358.png

InfProd_2359.png

InfProd_2360.png

InfProd_2361.png

InfProd_2362.png

InfProd_2363.png

InfProd_2364.png

InfProd_2365.png

InfProd_2366.png

InfProd_2367.png

InfProd_2368.png

InfProd_2369.png

InfProd_2370.png

InfProd_2371.png

InfProd_2372.png

InfProd_2373.png

InfProd_2374.png

InfProd_2375.png

InfProd_2376.png

InfProd_2377.png

InfProd_2378.png

InfProd_2379.png

InfProd_2380.png

InfProd_2381.png

InfProd_2382.png

InfProd_2383.png

InfProd_2384.png

InfProd_2385.png

InfProd_2386.png

InfProd_2387.png

InfProd_2388.gif

InfProd_2389.png

InfProd_2390.png

InfProd_2391.gif

InfProd_2392.png

InfProd_2393.png

InfProd_2394.png

InfProd_2395.png

InfProd_2396.png

InfProd_2397.png

InfProd_2398.png

InfProd_2399.png

InfProd_2400.png

InfProd_2401.png

InfProd_2402.png

InfProd_2403.png

InfProd_2404.png

InfProd_2405.png

InfProd_2406.png

InfProd_2407.png

InfProd_2408.png

InfProd_2409.png

InfProd_2410.png

InfProd_2411.png

InfProd_2412.png

InfProd_2413.png

InfProd_2414.png

InfProd_2415.png

InfProd_2416.png

InfProd_2417.png

InfProd_2418.png

InfProd_2419.png

InfProd_2420.png

InfProd_2421.png

InfProd_2422.png

InfProd_2423.png

InfProd_2424.png

InfProd_2425.png

InfProd_2426.png

InfProd_2427.png

InfProd_2428.png

InfProd_2429.png

InfProd_2430.png

InfProd_2431.png

InfProd_2432.png

InfProd_2433.png

InfProd_2434.png

InfProd_2435.png

InfProd_2436.png

InfProd_2437.png

InfProd_2438.png

InfProd_2439.png

InfProd_2440.png

InfProd_2441.png

InfProd_2442.png

InfProd_2443.png

InfProd_2444.png

InfProd_2445.png

InfProd_2446.png

InfProd_2447.png

InfProd_2448.png

InfProd_2449.png

InfProd_2450.png

InfProd_2451.png

InfProd_2452.png

Next is the 'Fountain' function, plot it in the range of -50 < z < 10 with parameter values of a between -3 and 1 !

InfProd_2453.png

InfProd_2454.png

InfProd_2455.png

InfProd_2456.gif

InfProd_2457.png

InfProd_2458.png

InfProd_2459.png

InfProd_2460.png

InfProd_2461.png

InfProd_2462.png

InfProd_2463.png

InfProd_2464.png

InfProd_2465.png

InfProd_2466.png

InfProd_2467.png

InfProd_2468.png

InfProd_2469.png

InfProd_2470.png

InfProd_2471.png

InfProd_2472.png

InfProd_2473.png

InfProd_2474.png

InfProd_2475.png

InfProd_2476.png

InfProd_2477.png

InfProd_2478.png

InfProd_2479.png

InfProd_2480.png

InfProd_2481.png

InfProd_2482.png

InfProd_2483.png

InfProd_2484.png

InfProd_2485.png

InfProd_2486.png

InfProd_2487.png

InfProd_2488.png

InfProd_2489.png

some of the next series appear also in electrostatic problems, see
Vigren E.; Dieckmann A. ; Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges. Symmetry 2020, 12, 1040.

InfProd_2490.png

InfProd_2491.png

InfProd_2492.png

InfProd_2493.png

InfProd_2494.png

derived from above series :

InfProd_2495.png

InfProd_2496.png

Series over prime numbers :

InfProd_2497.png

InfProd_2498.png

InfProd_2499.png

Series of Inverse Tangents ( Arcustangent ) :   ( Back to Top )

(m = InverseEllipticNomeQ[q], K[m] = EllipticK[m], E[m] = EllipticE[m]), the appearing of n or n - 1 as summation stop index implies n ∈ N.

InfProd_2500.png

InfProd_2501.png

InfProd_2502.png

InfProd_2503.png

InfProd_2504.png

InfProd_2505.png

InfProd_2506.png

InfProd_2507.png

InfProd_2508.png

InfProd_2509.png

InfProd_2510.png

InfProd_2511.png

InfProd_2512.png

InfProd_2513.png

InfProd_2514.png

InfProd_2515.png

InfProd_2516.png

InfProd_2517.png

InfProd_2518.png

InfProd_2519.png

InfProd_2520.gif

InfProd_2521.png

InfProd_2522.png

InfProd_2523.png

LogGamma[z] is used, because it has a simpler branch strucure than Log[Gamma[z]] and avoids many discontinuities.

InfProd_2524.png

InfProd_2525.png

InfProd_2526.png

InfProd_2527.png

InfProd_2528.png

InfProd_2529.png

InfProd_2530.png

InfProd_2531.png

InfProd_2532.png

InfProd_2533.png

InfProd_2534.png

InfProd_2535.png

InfProd_2536.png

InfProd_2537.png

InfProd_2538.png

InfProd_2539.png

InfProd_2540.png

InfProd_2541.png

InfProd_2542.png

InfProd_2543.png

InfProd_2544.png

InfProd_2545.png

InfProd_2546.png

InfProd_2547.png

InfProd_2548.png

InfProd_2549.png

InfProd_2550.png

InfProd_2551.png

InfProd_2552.png

InfProd_2553.png

InfProd_2554.png

InfProd_2555.png

InfProd_2556.png

InfProd_2557.png

InfProd_2558.png

InfProd_2559.png

InfProd_2560.png

InfProd_2561.png

InfProd_2562.png

InfProd_2563.png

InfProd_2564.png

InfProd_2565.png

InfProd_2566.png

InfProd_2567.png

InfProd_2568.png

InfProd_2569.png

InfProd_2570.png

InfProd_2571.png

InfProd_2572.png

Some ArcTan Identities :

InfProd_2573.gif

InfProd_2574.png

Series of Bessel Functions :   ( Back to Top )

InfProd_2575.png

InfProd_2576.png

InfProd_2577.png

InfProd_2578.png

InfProd_2579.png

InfProd_2580.png

InfProd_2581.png

InfProd_2582.png

InfProd_2583.png

InfProd_2584.png

InfProd_2585.png

InfProd_2586.png

InfProd_2587.png

InfProd_2588.png

InfProd_2589.png

InfProd_2590.png

InfProd_2591.png

InfProd_2592.png

InfProd_2593.png

InfProd_2594.png

InfProd_2595.png

InfProd_2596.png

InfProd_2597.png

InfProd_2598.png

InfProd_2599.png

InfProd_2600.png

InfProd_2601.png

InfProd_2602.png

InfProd_2603.png

InfProd_2604.png

InfProd_2605.png

InfProd_2606.png

InfProd_2607.png

InfProd_2608.png

InfProd_2609.png

InfProd_2610.png

InfProd_2611.png

InfProd_2612.png

InfProd_2613.png

InfProd_2614.png

InfProd_2615.png

InfProd_2616.png

InfProd_2617.png

InfProd_2618.png

InfProd_2619.png

InfProd_2620.png

InfProd_2621.png

InfProd_2622.png

InfProd_2623.png

InfProd_2624.png

Set a = 0 to get rid of the cos...

InfProd_2625.png

InfProd_2626.png

For numerical tests replace every ∞ in the results with the same (large enough) number.

InfProd_2627.png

InfProd_2628.png

InfProd_2629.png

InfProd_2630.png

InfProd_2631.png

InfProd_2632.png

InfProd_2633.png

InfProd_2634.png

InfProd_2635.png

InfProd_2636.png

InfProd_2637.png

InfProd_2638.png

InfProd_2639.png

InfProd_2640.png

InfProd_2641.png

InfProd_2642.png

InfProd_2643.png

InfProd_2644.png

InfProd_2645.png

InfProd_2646.png

InfProd_2647.png

InfProd_2648.png

InfProd_2649.png

InfProd_2650.png

InfProd_2651.png

InfProd_2652.png

InfProd_2653.png

InfProd_2654.png

InfProd_2655.png

InfProd_2656.png

InfProd_2657.png

InfProd_2658.png

InfProd_2659.png

InfProd_2660.png

InfProd_2661.png

InfProd_2662.png

InfProd_2663.png

InfProd_2664.png

InfProd_2665.png

InfProd_2666.png

InfProd_2667.png

InfProd_2668.png

InfProd_2669.png

InfProd_2670.png

InfProd_2671.png

InfProd_2672.png

InfProd_2673.png

Cases of Neumann' s addition theorem :

InfProd_2674.png

InfProd_2675.png

Cases of Graf' s addition theorem :

InfProd_2676.png

InfProd_2677.png

InfProd_2678.png

InfProd_2679.png

InfProd_2680.png

InfProd_2681.png

InfProd_2682.png

InfProd_2683.png

InfProd_2684.png

InfProd_2685.png

InfProd_2686.png

InfProd_2687.png

InfProd_2688.png

InfProd_2689.gif

InfProd_2690.png

InfProd_2691.png

InfProd_2692.png

InfProd_2693.png

InfProd_2694.png

InfProd_2695.png

Series of Legendre Polynomials :   ( Back to Top )

InfProd_2696.png

InfProd_2697.png

InfProd_2698.png

InfProd_2699.png

InfProd_2700.png

InfProd_2701.png

InfProd_2702.png

InfProd_2703.png

InfProd_2704.png

InfProd_2705.png

InfProd_2706.png

InfProd_2707.png

InfProd_2708.png

InfProd_2709.png

InfProd_2710.png

InfProd_2711.png

InfProd_2712.png

InfProd_2713.png

InfProd_2714.png

InfProd_2715.png

InfProd_2716.png

InfProd_2717.png

InfProd_2718.png

InfProd_2719.png

InfProd_2720.png

Laguerre Polynomials:

InfProd_2721.png

InfProd_2722.png

Laguerre Polynomials with negative Index

InfProd_2723.png

Series of Jacobi Polynomials :

InfProd_2724.png

InfProd_2725.png

Series of Hermite Polynomials :

InfProd_2726.png

InfProd_2727.png

InfProd_2728.png

InfProd_2729.png

InfProd_2730.png

InfProd_2731.png

Series of Zeta, PolyGamma, PolyLog and related :   ( Back to Top )

InfProd_2732.png

InfProd_2733.png

InfProd_2734.png

InfProd_2735.png

InfProd_2736.png

InfProd_2737.png

InfProd_2738.png

InfProd_2739.png

InfProd_2740.png

InfProd_2741.png

InfProd_2742.png

InfProd_2743.png

InfProd_2744.png

InfProd_2745.png

InfProd_2746.png

InfProd_2747.png

InfProd_2748.png

InfProd_2749.png

InfProd_2750.png

InfProd_2751.png

InfProd_2752.png

InfProd_2753.png

InfProd_2754.png

InfProd_2755.png

InfProd_2756.png

InfProd_2757.png

InfProd_2758.png

InfProd_2759.png

InfProd_2760.png

InfProd_2761.png

InfProd_2762.png

InfProd_2763.png

InfProd_2764.png

InfProd_2765.png

InfProd_2766.png

InfProd_2767.png

InfProd_2768.png

InfProd_2769.png

InfProd_2770.png

InfProd_2771.png

InfProd_2772.png

InfProd_2773.png

InfProd_2774.png

InfProd_2775.png

InfProd_2776.png

InfProd_2777.png

InfProd_2778.png

InfProd_2779.png

InfProd_2780.png

InfProd_2781.gif

Special Values of Zeta :

InfProd_2782.png

InfProd_2783.png

InfProd_2784.png

InfProd_2785.png

InfProd_2786.gif

InfProd_2787.png

InfProd_2788.png

PolyGamma :

InfProd_2789.png

InfProd_2790.png

InfProd_2791.png

InfProd_2792.png

InfProd_2793.png

InfProd_2794.png

InfProd_2795.png

InfProd_2796.png

InfProd_2797.png

InfProd_2798.png

InfProd_2799.png

InfProd_2800.png

InfProd_2801.png

InfProd_2802.png

InfProd_2803.png

InfProd_2804.png

InfProd_2805.png

InfProd_2806.png

InfProd_2807.png

InfProd_2808.png

InfProd_2809.png

InfProd_2810.png

InfProd_2811.png

InfProd_2812.png

InfProd_2813.png

InfProd_2814.png

InfProd_2815.gif

PolyLog and  LerchPhi :

InfProd_2816.png

InfProd_2817.png

InfProd_2818.png

InfProd_2819.png

InfProd_2820.png

InfProd_2821.png

InfProd_2822.png

InfProd_2823.png

InfProd_2824.png

InfProd_2825.png

InfProd_2826.png

InfProd_2827.png

InfProd_2828.png

InfProd_2829.png

InfProd_2830.gif

InfProd_2831.png

InfProd_2832.png

InfProd_2833.png

InfProd_2834.png

InfProd_2835.png

InfProd_2836.png

InfProd_2837.png

InfProd_2838.png

InfProd_2839.png

InfProd_2840.png

InfProd_2841.png

InfProd_2842.png

InfProd_2843.png

InfProd_2844.png

InfProd_2845.png

InfProd_2846.png

InfProd_2847.png

InfProd_2848.png

InfProd_2849.png

InfProd_2850.png

InfProd_2851.png

InfProd_2852.png

InfProd_2853.png

InfProd_2854.png

InfProd_2855.png

InfProd_2856.png

InfProd_2857.png

InfProd_2858.png

InfProd_2859.png

InfProd_2860.png

InfProd_2861.png

InfProd_2862.png

InfProd_2863.png

InfProd_2864.png

Special Values of PolyLog and LerchPhi :

InfProd_2865.png

InfProd_2866.png

InfProd_2867.png

InfProd_2868.png

InfProd_2869.png

InfProd_2870.png

InfProd_2871.png

InfProd_2872.png

InfProd_2873.png

InfProd_2874.png

InfProd_2875.png

InfProd_2876.png

InfProd_2877.png

InfProd_2878.png

InfProd_2879.png

InfProd_2880.gif

InfProd_2881.gif

Series of Beta Functions :   ( Back to Top )

Recurrence relation : Beta[x, a + 1, b] + Beta[x, a, b + 1] = Beta[x, a, b];

InfProd_2882.png

InfProd_2883.png

InfProd_2884.png

InfProd_2885.png

InfProd_2886.png

InfProd_2887.png

InfProd_2888.png

InfProd_2889.png

InfProd_2890.png

InfProd_2891.png

InfProd_2892.png

InfProd_2893.png

InfProd_2894.png

InfProd_2895.png

InfProd_2896.png

InfProd_2897.png

InfProd_2898.png

InfProd_2899.png

InfProd_2900.png

InfProd_2901.png

InfProd_2902.png

InfProd_2903.png

InfProd_2904.png

InfProd_2905.png

InfProd_2906.png

InfProd_2907.png

InfProd_2908.png

InfProd_2909.png

InfProd_2910.png

InfProd_2911.png

InfProd_2912.png

InfProd_2913.png

InfProd_2914.png

InfProd_2915.png

InfProd_2916.png

InfProd_2917.png

Special values of InverseBetaRegularized :

InfProd_2918.png

Series of Gamma Functions :   ( Back to Top )

InfProd_2919.png

Dougall' s Formula :

InfProd_2920.png

InfProd_2921.png

InfProd_2922.png

InfProd_2923.png

InfProd_2924.gif

InfProd_2925.png

InfProd_2926.png

InfProd_2927.png

InfProd_2928.png

InfProd_2929.png

InfProd_2930.png

InfProd_2931.png

InfProd_2932.png

InfProd_2933.png

InfProd_2934.png

InfProd_2935.png

InfProd_2936.png

InfProd_2937.png

InfProd_2938.png

InfProd_2939.png

note the offset of 1/2 that appears in the result of the second series above if a is set to 1.

InfProd_2940.png

InfProd_2941.png

InfProd_2942.png

InfProd_2943.png

InfProd_2944.png

InfProd_2945.png

InfProd_2946.png

The real part of the following Gamma series doesn' t converge :

InfProd_2947.png

InfProd_2948.png

InfProd_2949.png

InfProd_2950.png

InfProd_2951.png

InfProd_2952.png

InfProd_2953.png

Gamma Identities :

InfProd_2954.png

InfProd_2955.png

Special value of Gamma :

InfProd_2956.png

Special values of InverseGammaRegularized :

InfProd_2957.png

Series involving HarmonicNumber : ( Back To Top )

InfProd_2958.png

InfProd_2959.png

InfProd_2960.png

InfProd_2961.png

InfProd_2962.png

InfProd_2963.png

InfProd_2964.png

InfProd_2965.png

InfProd_2966.png

InfProd_2967.png

InfProd_2968.png

InfProd_2969.png

InfProd_2970.png

InfProd_2971.png

InfProd_2972.png

InfProd_2973.png

InfProd_2974.png

InfProd_2975.png

InfProd_2976.png

InfProd_2977.png

InfProd_2978.png

InfProd_2979.png

InfProd_2980.png

For n = 1 to 10    InfProd_2981.png is:

1: InfProd_2982.png -0.58224053
2: InfProd_2983.png -0.90797054
3: InfProd_2984.png -1.13055188
4: InfProd_2985.png -1.29927612
5: InfProd_2986.png -1.43505814
6: InfProd_2987.png -1.54863772
7: InfProd_2988.png -1.64624639
8: InfProd_2989.png -1.73181782
9: InfProd_2990.png -1.80799286
10: InfProd_2991.png -1.87662974

InfProd_2992.png

InfProd_2993.png

For n = 1 to 4     InfProd_2994.png is:

1: InfProd_2995.png 2.40411381
2: InfProd_2996.png 3.30565648
3: InfProd_2997.png 3.88459579
4: InfProd_2998.png 4.31204500

InfProd_2999.png

InfProd_3000.png

For n = 1 to 3     InfProd_3001.png is:

1: InfProd_3002.png -0.7512856
2: InfProd_3003.png -1.1496340
3: InfProd_3004.png -1.4185815

InfProd_3005.png

InfProd_3006.png

InfProd_3007.png

InfProd_3008.png

InfProd_3009.png

InfProd_3010.png

InfProd_3011.png

InfProd_3012.png

InfProd_3013.png

InfProd_3014.png

InfProd_3015.png

InfProd_3016.png

InfProd_3017.png

InfProd_3018.png

InfProd_3019.png

InfProd_3020.png

InfProd_3021.png

InfProd_3022.png

InfProd_3023.png

InfProd_3024.png

InfProd_3025.png

The notation in the Exponent of the hypergeometric function says : take the derivative with respect to the seventh argument and then give it the value 1 - a or 1 + a.

Series involving Hypergeometric Functions : ( Back to Top )

InfProd_3026.png

InfProd_3027.png

InfProd_3028.png

InfProd_3029.png

InfProd_3030.png

InfProd_3031.png

InfProd_3032.png

InfProd_3033.png

InfProd_3034.png

InfProd_3035.png

InfProd_3036.png

InfProd_3037.png

InfProd_3038.png

InfProd_3039.png

InfProd_3040.png

InfProd_3041.png

InfProd_3042.png

InfProd_3043.png

InfProd_3044.png

InfProd_3045.png

InfProd_3046.png

InfProd_3047.png

InfProd_3048.png

InfProd_3049.png

InfProd_3050.png

InfProd_3051.png

InfProd_3052.png

InfProd_3053.png

InfProd_3054.png

InfProd_3055.png

InfProd_3056.png

InfProd_3057.png

InfProd_3058.png

InfProd_3059.png

InfProd_3060.png

InfProd_3061.png

InfProd_3062.png

InfProd_3063.png

InfProd_3064.png

InfProd_3065.png

InfProd_3066.png

InfProd_3067.png

InfProd_3068.png

InfProd_3069.png

InfProd_3070.png

InfProd_3071.png

InfProd_3072.png

InfProd_3073.png

InfProd_3074.png

InfProd_3075.png

InfProd_3076.png

InfProd_3077.png

InfProd_3078.png

InfProd_3079.png

InfProd_3080.png

InfProd_3081.png

InfProd_3082.png

InfProd_3083.png

InfProd_3084.png

Some Limits :   ( Back to Top )

InfProd_3085.png

InfProd_3086.png

InfProd_3087.png

InfProd_3088.png

InfProd_3089.png

InfProd_3090.png

InfProd_3091.png

InfProd_3092.png

A few Integrals :   ( Back to Top )

InfProd_3093.png

Substitute  InfProd_3094.png   and the Feynman - Hibbs Integral

InfProd_3095.png

InfProd_3096.png

and derivatives :

InfProd_3097.png is the mth derivative with respect to a :

InfProd_3098.png

InfProd_3099.png

InfProd_3100.png

InfProd_3101.png

Iterated Expressions  ( Tetration ) :   ( Back to Top )

InfProd_3102.gif

InfProd_3103.png

InfProd_3104.gif

InfProd_3105.png

InfProd_3106.png

InfProd_3107.png

The above function f[x] = - ProductLog[-Log[x]] / Log[x] has a special 'swapping' symmetry of basis and exponent in its argument: InfProd_3108.png
f[x] is not defined beyond the maximum of its inverse function InfProd_3109.png, namely  InfProd_3110.png< x, so with this symmetry it is plausible that the exponential tower
doesn't converge for x < InfProd_3111.png as well, where it shows a kind of bifurcation.

Some Properties of ProductLog, LerchPhi and PolyLog   ( Back to Top )

For 1/e ≤ x    is ProductLog[ x Log[ x ]]        =    Log[ x ] .
For 0 ≤ x ≤ e    is ProductLog[ - Log[ x ] / x]    = - Log[ x ] .
For 0 ≤ x        is Log[ ProductLog[ x ] ]        =   Log[ x ] - ProductLog[ x ] .

InfProd_3112.png

InfProd_3113.png

InfProd_3114.png

InfProd_3115.png

1
1 2
1 8 6
1 22 58 24

InfProd_3116.png

For purely imaginary arguments (x ∈ R) the complex decomposition of LerchPhi is :

InfProd_3117.png

These carry over with a = 0 to PolyLog :

InfProd_3118.png

InfProd_3119.png

The imaginary part of LerchPhi[x, s, a] with 1 ≤ x ∈ R is given by :

InfProd_3120.png

And with a = 0 follows the imaginary part of PolyLog[ s, x] :

InfProd_3121.png

The complex decomposition of  InfProd_3122.png with 1 ≤ x ∈ R and 0 ≤ {b, s} ∈ N into real and imaginary part can be obtained by the following expression :

InfProd_3123.png

explicitly for low s and b = 2 :

InfProd_3124.png InfProd_3125.png
InfProd_3126.png InfProd_3127.png
InfProd_3128.png InfProd_3129.png
InfProd_3130.png InfProd_3131.png
InfProd_3132.png InfProd_3133.png
InfProd_3134.png InfProd_3135.png

For all z ∈ C not on the real axis in ( -∞ < z < 1) and 0 ≤ {b, s} ∈ N the following inversion identity holds
(the If statement inserts a '+' in case of an imaginary part of z larger than zero, a '-' in all other cases) :

InfProd_3136.png

The real part of  InfProd_3137.pngwith 1 ≤ x ∈ R is also given by

InfProd_3138.png

For (b ∈ N) is

InfProd_3139.png

The real and imaginary parts of LerchPhi[ InfProd_3140.png, 2, 1/2 ] (on the unit circle) are

InfProd_3141.png

With Clausen type functions for LerchPhi defined as

InfProd_3142.png

InfProd_3143.png

(0 < s ∈ Integer, 0 ≤ θ ≤ 2π, the even CLi and the odd SLi are expressible through Euler Polynomials),
the real and imaginary parts of InfProd_3144.pngInfProd_3145.png (on the unit circle) are

InfProd_3146.png

the expressions for InfProd_3147.png with lowest s being

InfProd_3148.png InfProd_3149.png
InfProd_3150.png InfProd_3151.png
InfProd_3152.png InfProd_3153.png
InfProd_3154.png InfProd_3155.png
InfProd_3156.png InfProd_3157.png
InfProd_3158.png InfProd_3159.png

The above polynomials in a make nice approximations to trigonometric functions, getting better with increasing s.
The first non polynomial partnerfunctions are found to be

InfProd_3160.png

The function InfProd_3161.png has an interesting derivative :

InfProd_3162.png

that means the lower CLi and SLi are essentially derivatives of the higher ones.

With the LerchPhi index n being a negative Integer the function appears as a rational function :

InfProd_3163.png

With the PolyLog index being a negative Integer the function appears as a rational function :

InfProd_3164.png

With Clausen type functions defined as

InfProd_3165.png

InfProd_3166.png

(0 < s ∈ Integer, 0 ≤ θ ≤ 2π,  the even Ci and the odd Si are expressible through Bernoulli Polynomials),
the real and imaginary parts of InfProd_3167.png (on the unit circle) are

InfProd_3168.png

the expressions for InfProd_3169.png with lowest s being

InfProd_3170.png

The above polynomials in a make nice approximations to trigonometric functions, getting better with increasing s :

InfProd_3171.png

As before the derivative InfProd_3172.pngInfProd_3173.pngis InfProd_3174.png with lowered index.
The first non polynomial partnerfunctions are found to be

InfProd_3175.png

The complex decomposition of  PolyLog[s, x] with 1 ≤ x ∈ R and 0 ≤ s ∈ N can be obtained by the following expression :

InfProd_3176.png

explicitly for low s :

InfProd_3177.png InfProd_3178.png
InfProd_3179.png InfProd_3180.png
InfProd_3181.png InfProd_3182.png
InfProd_3183.png InfProd_3184.png
InfProd_3185.png InfProd_3186.png
InfProd_3187.png InfProd_3188.png
InfProd_3189.png InfProd_3190.png

For real x < 1  is :

InfProd_3191.png

InfProd_3192.png Log[x]-Log[1-x]-i π
InfProd_3193.png InfProd_3194.png
InfProd_3195.png InfProd_3196.png
InfProd_3197.png InfProd_3198.png
InfProd_3199.png InfProd_3200.png
InfProd_3201.png InfProd_3202.png

For all z ∈ C and not on the real axis in ( 0 ≤ z < 1 ) and 0 ≤  s ∈ N the following inversion identity holds :

InfProd_3203.png

LerchPhi and PolyLog display a similar (alternating with s) scheme in their real and imaginary parts :

InfProd_3204.png

InfProd_3205.png

InfProd_3206.png

InfProd_3207.png

InfProd_3208.png

InfProd_3209.png

InfProd_3210.png

InfProd_3211.png

The lowest Bernoulli and Euler Polynomials are

BernoulliB EulerE
InfProd_3212.png InfProd_3213.png InfProd_3214.png
InfProd_3215.png InfProd_3216.png InfProd_3217.png
InfProd_3218.png InfProd_3219.png InfProd_3220.png
InfProd_3221.png InfProd_3222.png InfProd_3223.png
InfProd_3224.png InfProd_3225.png InfProd_3226.png
InfProd_3227.png InfProd_3228.png InfProd_3229.png

They are symmetric or antisymmetric (depending on n) with respect to x = 1/2 :

InfProd_3230.png

Connection to Bernoulli and Euler numbers :

InfProd_3231.png

Clausen functions and integral :

InfProd_3232.png

InfProd_3233.png

InfProd_3234.png

InfProd_3235.png

Contributors :

(R1) Udo Ausserlechner, Infineon, per email

(R2) Professor Don Zagier, MPI für Mathematik, Bonn

(R3) Stephen, per email

(R4) Erik Vigren, IRF, Uppsala, per email

Created with the Wolfram Language      Download Page    Indefinite Integrals     Definite Integrals