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];
sl[x]  cl[x] and ϖ denote the lemniscate functions and constant  InfProd_7.png ;
the notation Σ’ means that the divergent term in multiple sums is excluded.
Some of the products 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_8.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_9.png

InfProd_10.png

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InfProd_12.png

InfProd_13.png

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InfProd_58.png

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Products of two Gammas :

InfProd_60.png

InfProd_61.png

InfProd_62.png

Partial Fraction Decompositions :

InfProd_63.png

General expression :

InfProd_64.png

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

Order 2:

with n = 2 :

InfProd_65.png

most common  case (a quadratic binomial)

with  InfProd_66.png  and  InfProd_67.png:

InfProd_68.png

Decomposition of the general quadratic trinomial applying the shorthands

InfProd_69.png

gives

InfProd_70.png

Order 3:

with n = 3  :

InfProd_71.png

With 4 abbreviations

InfProd_72.gif

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

InfProd_73.png

Order 4:

A simple one :

InfProd_74.png

And at last using these 9  subexpressions

InfProd_75.gif

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

InfProd_76.png

  InfProd_77.png as simple function of k :

Decomposition of  finite products into power series

InfProd_78.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) :

InfProd_79.png

InfProd_80.png

InfProd_81.png

InfProd_82.png

InfProd_83.png

InfProd_84.png

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Special cases with m = 0 :

InfProd_86.png

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InfProd_89.png

InfProd_90.png

q - Product (0 < q < 1) :

InfProd_91.png

Two kinds of decomposition of the same product :

InfProd_92.gif

With

InfProd_93.png

(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_94.png

If f[k] = InfProd_95.png  then

InfProd_96.png

or if f[k] = InfProd_97.png  then

InfProd_98.png

InfProd_99.png

InfProd_100.png

InfProd_101.png

InfProd_102.png

More Products :

The maximum of the next function is found at InfProd_103.png | f InfProd_104.png)) =  InfProd_105.png,  its zeros on the positive (negative) axis are the odd (even) Integers and in general f[z] = f[-z-1] :

InfProd_106.png

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Euler’s product :

InfProd_128.png

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The next product approximates a Gauss function InfProd_137.png with InfProd_138.png for a large value of m:

InfProd_139.gif

InfProd_140.png

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Products of trig functions :

InfProd_147.png

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Products containing factors built from all primes p:

InfProd_192.png

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InfProd_194.png

Products involving Theta Functions    ( Back to Top )

InfProd_195.png is shorthand for EllipticTheta[n, z, q] and InfProd_196.png means EllipticThetaPrime[m, z, q].

InfProd_197.png

( The above double product numerically converges best if k ≫ n. )

Series and Product Representations :

InfProd_198.png

InfProd_199.png

InfProd_200.png

InfProd_201.png

InfProd_202.png

With  InfProd_203.png[ 0 , q ] a few relations between the theta functions are

InfProd_204.gif

InfProd_205.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_213.png

Approximation of quotients (0.4 < λ) :

InfProd_214.gif

InfProd_215.png

InfProd_216.png

Partial differential equation :

InfProd_217.png

EllipticThetas with imaginary argument :

InfProd_218.png

With z → 0 we get

InfProd_219.png

Half Lambda :

InfProd_220.png

InfProd_221.png

InfProd_222.png

InfProd_223.png

InfProd_224.png

InfProd_225.png

Double Lambda :

InfProd_226.png

InfProd_227.png

Other relations :

InfProd_228.png

Square and square root of q :

InfProd_229.png

InfProd_230.png

Double Argument  (Landen), -  see above for double  InfProd_231.png)  and half  InfProd_232.pngLambda - :

InfProd_233.png

InfProd_234.png

InfProd_235.png

InfProd_236.png

Half Argument :

InfProd_237.png

InfProd_238.png

InfProd_239.png

InfProd_240.png

InfProd_241.png

InfProd_242.png

Series representation of ratios of theta functions :

InfProd_243.png

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

InfProd_244.png

InfProd_245.png

Double product representation of the single theta functions :

InfProd_246.png

InfProd_247.png

InfProd_248.png

InfProd_249.png

If the product over k is carried out first we get products with Tanh and Coth :

InfProd_250.png

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The theta functions may be expressed through each other :

InfProd_258.png

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

InfProd_259.png

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

InfProd_260.png

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Trigonometric and hyperbolic Products :

InfProd_282.png

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InfProd_295.png

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

InfProd_298.png

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InfProd_321.png

InfProd_322.png

q - Products :

In the following is ( 0 < q < 1 ) and InfProd_323.png[ 0 , q ] ,   (InfProd_324.png[ 0 , q ] =InfProd_325.png[ 0 , - q ] ) :

InfProd_326.png

InfProd_327.png

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

InfProd_328.png

InfProd_329.gif

InfProd_330.png

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

InfProd_360.png

InfProd_361.png

InfProd_362.png

InfProd_363.png

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

InfProd_364.gif

InfProd_365.png

InfProd_366.png

InfProd_367.png

InfProd_368.pngInfProd_369.png and InfProd_370.png can be expressed through m[q] , K[m[q]] and E[m[q]] :

InfProd_371.png

and similarly :

InfProd_372.png

and :

InfProd_373.png

and from combining the above like :

InfProd_374.png

we get :

InfProd_375.png

as q is getting larger than InfProd_376.png the branch cut of K and E is crossed, so the continuous and smooth complex functions are built from two parts :

InfProd_377.png

or turned the other way round :

InfProd_378.png

InfProd_379.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 i 2 π

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

InfProd_380.png InfProd_381.png InfProd_382.png InfProd_383.png
InfProd_384.png InfProd_385.png InfProd_386.png InfProd_387.png
InfProd_388.png InfProd_389.png InfProd_390.png InfProd_391.png
InfProd_392.png InfProd_393.png InfProd_394.png InfProd_395.png
InfProd_396.png InfProd_397.png InfProd_398.png InfProd_399.png
InfProd_400.png InfProd_401.png InfProd_402.png InfProd_403.png
InfProd_404.png InfProd_405.png InfProd_406.png InfProd_407.png
InfProd_408.png InfProd_409.png InfProd_410.png InfProd_411.png

Series expansion of InverseEllipticNomeQ :

InfProd_412.png

InfProd_413.png

InfProd_414.png

Connection of InverseEllipticNomeQ to ModularLambda :

InfProd_415.png

InfProd_416.png

InfProd_417.png

InfProd_418.gif

Series expansion of EllipticNomeQ :

InfProd_419.png

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Specific Values :

InfProd_421.png

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InfProd_442.png and InfProd_443.png are Ramanujans g functions, m = InverseEllipticNomeQ[InfProd_444.png]  (for each n ∈ Integer  the even g and the odd G seem to show a somewhat simpler structure than their counterparts) :

InfProd_445.png

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products with q = InfProd_458.png :

m = InverseEllipticNomeQ[InfProd_459.png] :

InfProd_460.png

InfProd_461.png

InfProd_462.png

InfProd_463.png

special cases :

InfProd_464.png

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InfProd_468.png

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Theta Functions, specific values :

InfProd_492.png

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InfProd_494.png

InfProd_495.gif

InfProd_496.gif

InfProd_497.png

InfProd_498.gif

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InfProd_522.gif

InfProd_523.png

Beauty meets well-tempered music…;-) ↓

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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_546.png means InfProd_547.png.

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

InfProd_548.png

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

InfProd_559.png

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Series involving exponentials :

InfProd_572.png

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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_589.png

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InfProd_614.png

Series involving InfProd_615.png :

InfProd_616.gif

InfProd_617.png

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InfProd_619.png

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InfProd_644.png

InfProd_645.gif

InfProd_646.png

InfProd_647.gif

InfProd_648.png

InfProd_649.png

Series of Stirling numbers :

InfProd_650.png

InfProd_651.png

InfProd_652.png

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

InfProd_653.png

InfProd_654.png

Series of trigonometric functions :

InfProd_655.png

InfProd_656.png

InfProd_657.png

InfProd_658.png

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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

InfProd_689.png

InfProd_690.png

InfProd_691.png

InfProd_692.png

InfProd_693.png

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.png

InfProd_721.png

InfProd_722.png

InfProd_723.png

InfProd_724.png

InfProd_725.png

InfProd_726.png

InfProd_727.png

InfProd_728.png

InfProd_729.png

InfProd_730.png

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

InfProd_731.png

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.gif

InfProd_747.png

InfProd_748.png

InfProd_749.png

InfProd_750.png

InfProd_751.png

InfProd_752.png

InfProd_753.png

InfProd_754.png

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.gif

InfProd_784.png

InfProd_785.png

InfProd_786.png

Amazing identities connecting trigonometric and lemniscate functions:

InfProd_787.png

InfProd_788.png

Special values of trigonometric functions :

Euler :

InfProd_789.png

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

InfProd_790.png

1
InfProd_791.png InfProd_792.png
InfProd_793.png 1 InfProd_794.png
InfProd_795.png InfProd_796.png InfProd_797.png InfProd_798.png
InfProd_799.png InfProd_800.png 1 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 1 InfProd_812.png InfProd_813.png InfProd_814.png

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

InfProd_815.png

0
InfProd_816.png InfProd_817.png
InfProd_818.png 0 InfProd_819.png
InfProd_820.png InfProd_821.png InfProd_822.png InfProd_823.png
InfProd_824.png InfProd_825.png 0 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.png InfProd_835.png InfProd_836.png 0 InfProd_837.png InfProd_838.png InfProd_839.png

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

InfProd_840.png

ComplexInfinity
InfProd_841.png InfProd_842.png
1 ComplexInfinity -1
InfProd_843.png InfProd_844.png InfProd_845.png InfProd_846.png
InfProd_847.png InfProd_848.png ComplexInfinity 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 1 InfProd_858.png ComplexInfinity InfProd_859.png -1 InfProd_860.png

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

InfProd_861.png

0
InfProd_862.png InfProd_863.png
1 0 -1
InfProd_864.png InfProd_865.png InfProd_866.png InfProd_867.png
InfProd_868.png InfProd_869.png 0 InfProd_870.png InfProd_871.png
InfProd_872.png InfProd_873.png InfProd_874.png InfProd_875.png InfProd_876.png InfProd_877.png
InfProd_878.png 1 InfProd_879.png 0 InfProd_880.png -1 InfProd_881.png

Some (special) special values :

InfProd_882.png InfProd_883.png
InfProd_884.png InfProd_885.png
InfProd_886.png InfProd_887.png

InfProd_888.png

InfProd_889.png InfProd_890.png
InfProd_891.png InfProd_892.png
InfProd_893.png InfProd_894.png

q - Series :   ( Back to Top )

with InfProd_895.png → Cosh[ k Log[ q ]] + Sinh[ k Log[ q ]] the following expressions can be transformed into sums of hyperbolic functions.

InfProd_896.gif

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.

InfProd_897.png

InfProd_898.png

InfProd_899.png

InfProd_900.png

There is a small stumble stone in the definition of LerchPhi in the neighbourhood of a = 0: LerchPhi[q, n, a] = InfProd_901.png, it changes for a = 0 abruptly to a different function InfProd_902.png.

InfProd_903.png

InfProd_904.png

InfProd_905.png

InfProd_906.png

InfProd_907.png

InfProd_908.png

InfProd_909.png

InfProd_910.png

InfProd_911.png

InfProd_912.png

InfProd_913.gif

InfProd_914.gif

InfProd_915.png

InfProd_916.png

InfProd_917.png

InfProd_918.png

InfProd_919.gif

InfProd_920.png

InfProd_921.png

InfProd_922.png

Using an identity from (R4) :

InfProd_923.png

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

InfProd_924.png

InfProd_925.png

InfProd_926.png

InfProd_927.png

InfProd_928.png

InfProd_929.png

InfProd_930.png

InfProd_931.png

InfProd_932.png

InfProd_933.png

InfProd_934.png

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

Lambert Type q Series:

InfProd_935.png

InfProd_936.png

InfProd_937.png

InfProd_938.gif

InfProd_939.png

InfProd_940.gif

InfProd_941.gif

InfProd_942.png

InfProd_943.png

InfProd_944.gif

InfProd_945.png

InfProd_946.gif

InfProd_947.gif

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

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

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

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

InfProd_1016.png

InfProd_1017.png

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

InfProd_1028.png

For 0.2 < q is in good approximation :

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

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

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

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

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

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

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

InfProd_1099.png

InfProd_1100.png

InfProd_1101.png

InfProd_1102.png

InfProd_1103.png

InfProd_1104.png

InfProd_1105.png

InfProd_1106.png

InfProd_1107.png

InfProd_1108.png

InfProd_1109.png

InfProd_1110.png

InfProd_1111.png

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

InfProd_1121.png

InfProd_1122.png

InfProd_1123.png

InfProd_1124.png

InfProd_1125.png

InfProd_1126.png

InfProd_1127.png

InfProd_1128.png

InfProd_1129.png

InfProd_1130.png

InfProd_1131.png

InfProd_1132.png

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

InfProd_1142.png

InfProd_1143.png

InfProd_1144.png

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.gif

InfProd_1158.png

InfProd_1159.png

InfProd_1160.png

InfProd_1161.png

InfProd_1162.png

InfProd_1163.png

InfProd_1164.png

InfProd_1165.png

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.png

InfProd_1178.png

InfProd_1179.png

other :

InfProd_1180.png

InfProd_1181.png

InfProd_1182.png

InfProd_1183.png

InfProd_1184.png

InfProd_1185.png

InfProd_1186.png

InfProd_1187.png

InfProd_1188.png

InfProd_1189.png

InfProd_1190.png

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

QFunction Identities :

InfProd_1201.png

InfProd_1202.png

InfProd_1203.png

InfProd_1204.png

InfProd_1205.png

InfProd_1206.png

InfProd_1207.png

InfProd_1208.png

InfProd_1209.png

InfProd_1210.png

InfProd_1211.gif

InfProd_1212.png

InfProd_1213.png

InfProd_1214.png

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.gif

InfProd_1253.png

With x ∈ Reals is   InfProd_1254.png

InfProd_1255.png
Real Part Imaginary Part
n=1: InfProd_1256.png 0
n=2: InfProd_1257.png InfProd_1258.png
n=3: InfProd_1259.png InfProd_1260.png
n=4: InfProd_1261.png InfProd_1262.png
n=5: InfProd_1263.png InfProd_1264.png
n=6: InfProd_1265.png InfProd_1266.png
n=7: InfProd_1267.png InfProd_1268.png

InfProd_1269.png

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

InfProd_1270.png

E[m] is EllipticE[m];

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

Series of Hyperbolic Functions:   ( Back to Top )

InfProd_1285.png

InfProd_1286.png

InfProd_1287.png

InfProd_1288.png

InfProd_1289.gif

InfProd_1290.gif

InfProd_1291.png

InfProd_1292.png

InfProd_1293.png

ϑ ‘’ represents the second derivative of ϑ (x,q) with respect to x, eg: InfProd_1294.png means InfProd_1295.png.

For numerical checks the finite lower negative index limit should be increased by 1 where the running index is shifted by - 1/2.

InfProd_1296.png

InfProd_1297.png

InfProd_1298.png

InfProd_1299.png

InfProd_1300.png

InfProd_1301.png

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

InfProd_1302.png

InfProd_1303.png

InfProd_1304.png

InfProd_1305.png

InfProd_1306.png

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

In the series marked with {*} the absolute value of the (negative) start index should be 1 lower than the stop index, in numerical use. Otherwise you will not get equality.

InfProd_1321.png

InfProd_1322.png

InfProd_1323.png

InfProd_1324.png

In the series marked with {*} the absolute value of the (negative) start index should be 1 lower than the stop index, in numerical use. Otherwise you will not get equality.

InfProd_1325.png

InfProd_1326.png

InfProd_1327.png

InfProd_1328.png

InfProd_1329.png

InfProd_1330.png

Hyperbolic series involving the lemniscate functions :

ϖ is the lemniscate constant :

InfProd_1331.png

InfProd_1332.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_1333.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_1334.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_1335.png needed to obtain the series for cl) :

InfProd_1336.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

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_1337.png

InfProd_1338.png

The 'four horsemen of the apocalypse':

InfProd_1339.png

InfProd_1340.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_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

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_1352.png

InfProd_1353.png

InfProd_1354.png

InfProd_1355.png

In the series marked with {*} the absolute value of the (negative) start index should be 1 lower than the stop index, in numerical use. Otherwise you will not get equality.

InfProd_1356.png

InfProd_1357.png

InfProd_1358.png

The QPolyGamma 'monsters' :

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

m = InverseEllipticNomeQ[InfProd_1368.png] :

InfProd_1369.png

InfProd_1370.png

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

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_1391.png

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

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

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

InfProd_1401.png

InfProd_1402.png

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

InfProd_1403.png

InfProd_1404.png

InfProd_1405.png

InfProd_1406.png

m = InverseEllipticNomeQ[InfProd_1407.png] :

InfProd_1408.png

InfProd_1409.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_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.png

InfProd_1431.png

InfProd_1432.png

InfProd_1433.png

InfProd_1434.png

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

m = InverseEllipticNomeQ[InfProd_1446.png] :

InfProd_1447.png

InfProd_1448.png

InfProd_1449.png

InfProd_1450.png

InfProd_1451.png

Some hyperbolic Identities :

InfProd_1452.png

InfProd_1453.png

InfProd_1454.gif

InfProd_1455.gif

Some Lemniscate Sine and Cosine Identities including derivative and integral:

Periods :

InfProd_1456.png

Dual sibling of the Pythagorean Identity InfProd_1457.png) :

InfProd_1458.png

Special values :

InfProd_1459.png

InfProd_1460.png

InfProd_1461.png

InfProd_1462.png

InfProd_1463.png

Argument addition formulae :

InfProd_1464.png

Imaginary, negative and double arguments :

InfProd_1465.png

Squares :

InfProd_1466.png

Derivatives and basic integrals :

InfProd_1467.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_1468.png (sl[x + dx] - sl[x])/dx = cl[x] (1 + InfProd_1469.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

One can verify the following identities using the imaginary and symmetry properties of sl and cl given above.

InfProd_1471.png

Complex properties :

InfProd_1472.png

Product approximations to the lemniscate functions (for 0 < m both better than InfProd_1473.png for all x, they  converge more than twice as fast as the already fast converging series given above in the section) :

InfProd_1474.png

Lemniscate functions as special cases  InfProd_1475.png of Jacobi elliptic functions  or Neville theta functions:

InfProd_1476.png

Series of CosIntegral:   ( Back to Top )

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

InfProd_1490.png

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

InfProd_1491.png

InfProd_1492.png

InfProd_1493.png

InfProd_1494.png

InfProd_1495.png

InfProd_1496.png

For even m  the right hand side reduces to

InfProd_1497.png

InfProd_1498.png

Higher multifactorials :

In the next four equations m designates the number of ! within the multifactorial InfProd_1499.png, γ[a, b] is the lower incomplete Gamma Function Γ[a, 0, b] = InfProd_1500.png .

InfProd_1501.png

InfProd_1502.png

InfProd_1503.png

InfProd_1504.png

InfProd_1505.png

Expressions for still higher orders of k may be obtained by applying InfProd_1506.png to both sides of the equation.

The general expression at x = 1 is, using recursionally defined coefficients  InfProd_1507.png and InfProd_1508.png (the factor after InfProd_1509.png shows the series result for n = 0) :

InfProd_1510.png

where the recursions

InfProd_1511.png

and

InfProd_1512.png

give the integers InfProd_1513.png and InfProd_1514.png, needed for the calculation of the sum above .

The coefficients obtained with low indices m, n (m 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_1515.png

InfProd_1516.png

The numbers InfProd_1517.png and InfProd_1518.png represent the Bell numbers BellB[n, 1] and the complementary Bell numbers BellB[n, -1].

InfProd_1519.png

InfProd_1520.png

InfProd_1521.png

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

InfProd_1522.png

or  may be calculated as higher derivatives of the exponential function : BellB[n, z] InfProd_1523.png  :

InfProd_1524.png

1 1 1 1
z z z -z
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

A completely crazy series :

InfProd_1548.png

where  the convergence limits for x were determined empirically as InfProd_1549.png  = 0.00027 InfProd_1550.png- 0.28 and InfProd_1551.png = somewhat above 0.8, depending on n, determined for a range of n = -1 to 20.

The expression in large brackets represents a polynomial in y and InfProd_1552.png of degree n with integer coefficients.

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

InfProd_1553.png

and appears in a special combinatorics problem :
It calculates the count of possible ways InfProd_1554.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 :

In[35]:=

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 balls in 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_1555.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, 2])  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_1556.png

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

InfProd_1557.png

InfProd_1558.png

Higher powers of the factorial in the denominator :

InfProd_1559.gif

Replace InfProd_1560.png with InfProd_1561.png and InfProd_1562.png  with InfProd_1563.png
as well as  InfProd_1564.png with InfProd_1565.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 possible imaginary contributions.

Expressions for still higher orders of k above may be obtained by applying (InfProd_1566.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_1567.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_1568.png

0 < j < m:

InfProd_1569.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_1570.pnginto sums of powers InfProd_1571.png less than InfProd_1572.png:

InfProd_1573.png

and the recursion for the coefficients c is given by :

InfProd_1574.png

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

InfProd_1575.png

For example with n = 6, m = 3 :

InfProd_1576.png

InfProd_1577.png

The InfProd_1578.pngs are then :

InfProd_1579.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_1580.png

InfProd_1581.png

so that the recursionally defined sum yields :

InfProd_1582.png

while Mathematica gives:

InfProd_1583.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_1584.png

The  highest exponent of InfProd_1585.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 a little easier, because all sums (0 ≤ n)InfProd_1586.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_1587.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_1588.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_1589.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_1590.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]

The series

InfProd_1591.png

converge very fast (the terms k > 3 contribute less than InfProd_1592.png) . For m = 3  the terms give :

InfProd_1593.png

With n = 0 the first sum and (InfProd_1594.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_1595.png

InfProd_1596.png

InfProd_1597.png

InfProd_1598.png

InfProd_1599.png

InfProd_1600.png

InfProd_1601.png

InfProd_1602.png

InfProd_1603.png

InfProd_1604.png

InfProd_1605.png

InfProd_1606.png

InfProd_1607.png

InfProd_1608.png

InfProd_1609.png

The notation  InfProd_1610.pngmeans that the divergent term 1/0 is excluded .

Borwein' s formula :

InfProd_1611.png

Benson' s formula:

InfProd_1612.png

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

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

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_1631.png

InfProd_1632.png

Double series involving lemniscate sine and cosine :

InfProd_1633.png

InfProd_1634.png

InfProd_1635.png

The last sum above shows 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.

The notation InfProd_1636.png means : you get the form of function g from the derivative (integral) of function f.

InfProd_1637.png

InfProd_1638.png

InfProd_1639.png

InfProd_1640.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_1641.png

InfProd_1642.png

InfProd_1643.png

InfProd_1644.png

InfProd_1645.png

InfProd_1646.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_1647.png

InfProd_1648.png

InfProd_1649.png

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

InfProd_1650.png

Double Series involving the Zeta function :

InfProd_1651.png

InfProd_1652.png

InfProd_1653.png

InfProd_1654.gif

Multiple Series involving the Zeta function :

InfProd_1655.gif

InfProd_1656.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_1657.png
Zeta[3] InfProd_1658.png
InfProd_1659.png InfProd_1660.png InfProd_1661.png
Zeta[5] 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

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_1671.png
Zeta[3] InfProd_1672.png
InfProd_1673.png Zeta[3] InfProd_1674.png
Zeta[5] InfProd_1675.png InfProd_1676.png InfProd_1677.png
InfProd_1678.png Zeta[5] InfProd_1679.png InfProd_1680.png InfProd_1681.png
Zeta[7] InfProd_1682.png InfProd_1683.png InfProd_1684.png InfProd_1685.png InfProd_1686.png

The notation InfProd_1687.pngmeans that the divergent term 1/0 is excluded :

InfProd_1688.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_1689.png 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 InfProd_1715.png InfProd_1716.png InfProd_1717.png

InfProd_1718.png

where the number of numerically equal summands  InfProd_1719.png) in the first sum is counted by the multiplicity factor in large brackets within the second sum;

InfProd_1720.png

s controls the number if integers 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_1721.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_1722.png InfProd_1723.png InfProd_1724.png InfProd_1725.png
InfProd_1726.png Log[2] InfProd_1727.png InfProd_1728.png InfProd_1729.png
InfProd_1730.png InfProd_1731.png InfProd_1732.png InfProd_1733.png 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

For a few values of s = 1 to 6 (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_1751.png) is determined by Binomial[k - 1, m - 1]:

InfProd_1752.png

-Log[2] 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 InfProd_1770.png InfProd_1771.png
InfProd_1772.png InfProd_1773.png InfProd_1774.png InfProd_1775.png InfProd_1776.png
InfProd_1777.png InfProd_1778.png InfProd_1779.png InfProd_1780.png InfProd_1781.png
InfProd_1782.png InfProd_1783.png InfProd_1784.png InfProd_1785.png InfProd_1786.png

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

InfProd_1787.png

For instance  (find an identity with start index 0, replace  kInfProd_1788.png ,  insert the Gammas, the j - 1 factorial  and sum over all i's)

InfProd_1789.png

For a series  InfProd_1790.png starting with index 1 it is a little more involved, because the first lattice summand will be InfProd_1791.png :

InfProd_1792.png

So the lattice version  (for example) of

InfProd_1793.png

InfProd_1794.png

Lattice q-sums :

InfProd_1795.png

InfProd_1796.png

InfProd_1797.png

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

InfProd_1798.png

where the summands characterized by  InfProd_1799.png = k occur with a certain multiplicity InfProd_1800.png given by

InfProd_1801.png

This tells the number of ways to express k as a sum of m integers InfProd_1802.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_1803.png

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

InfProd_1804.png

This identity can be used to successively get values of InfProd_1805.png. It  determines InfProd_1806.png at m = 3:

InfProd_1807.png

Expanding the sum over (k+2) shows InfProd_1808.png and InfProd_1809.png, and if it is assumed that InfProd_1810.png = 1/2 (regularization), then it follows that InfProd_1811.png = - 1/4.

For m = n + 2 the sum reads :

InfProd_1812.png

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

InfProd_1813.png

2+k 2+k
InfProd_1814.png InfProd_1815.png
InfProd_1816.png InfProd_1817.png
InfProd_1818.png InfProd_1819.png
InfProd_1820.png InfProd_1821.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_1822.pnglike:

InfProd_1823.png

InfProd_1824.png

Shown here are the InfProd_1825.png for n from 0 to 10 together with the results of corresponding Mathematica sums employing ' Abel' regularization as well as the symbolic result given at the start of the paragraph.

diverse Series :   ( Back to Top )

InfProd_1826.png

InfProd_1827.png

InfProd_1828.png

InfProd_1829.png

InfProd_1830.png

InfProd_1831.png

InfProd_1832.png

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

InfProd_1833.png

InfProd_1834.png

InfProd_1835.png

InfProd_1836.png

InfProd_1837.png

InfProd_1838.png

InfProd_1839.png

InfProd_1840.png

InfProd_1841.png

InfProd_1842.png

InfProd_1843.png

InfProd_1844.png

InfProd_1845.png

InfProd_1846.png

InfProd_1847.png

InfProd_1848.png

InfProd_1849.png

InfProd_1850.png

InfProd_1851.png

InfProd_1852.png

InfProd_1853.png

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.gif

InfProd_1867.png

InfProd_1868.png

InfProd_1869.png

InfProd_1870.png

InfProd_1871.png

InfProd_1872.png

Sum of the inverse m - gonal numbers :

InfProd_1873.png

InfProd_1874.png

Values of the series for the first m :

InfProd_1875.png

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

InfProd_1876.png

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

InfProd_1877.png

k - nomial triangles:

Generate the  k - nomial triangle  as coefficents of InfProd_1878.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_1879.gif

other series:

The sum  InfProd_1880.png  gives following results for some rational s = p/q :

InfProd_1881.png

InfProd_1882.png

InfProd_1883.png

InfProd_1884.png

InfProd_1885.png

InfProd_1886.png

InfProd_1887.png

InfProd_1888.png

InfProd_1889.png

InfProd_1890.png

InfProd_1891.png

InfProd_1892.png

InfProd_1893.png

InfProd_1894.png

InfProd_1895.png

InfProd_1896.png

InfProd_1897.png

InfProd_1898.png

InfProd_1899.png

InfProd_1900.png

InfProd_1901.png

InfProd_1902.png

InfProd_1903.png

InfProd_1904.png

InfProd_1905.png

InfProd_1906.png

InfProd_1907.png

InfProd_1908.png

InfProd_1909.png

InfProd_1910.png

This sum alternates between ± π  for z ∈ N :

InfProd_1911.png

In the following 4 expressions b =InfProd_1912.png :

InfProd_1913.png

InfProd_1914.png

InfProd_1915.png

InfProd_1916.png

InfProd_1917.png

InfProd_1918.png

InfProd_1919.png

InfProd_1920.png

InfProd_1921.png

InfProd_1922.png

InfProd_1923.png

The next three expressions contain s = InfProd_1924.png and t = 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

During a visit in London Leibniz was asked by Huygens to evaluate the sum InfProd_1938.png. He knew about partial fraction decomposition (see next line) and solved it. (Hirsch, der berühmte Herr Leibniz : eine Biographie)

InfProd_1939.png

InfProd_1940.png

InfProd_1941.png

InfProd_1942.png

The sum of  integer powers  of the inverse trigonal numbers :

InfProd_1943.png

InfProd_1944.png

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

InfProd_1964.png

InfProd_1965.png

InfProd_1966.png

InfProd_1967.png

InfProd_1968.png

InfProd_1969.png

InfProd_1970.png

InfProd_1971.png

InfProd_1972.png

InfProd_1973.png

InfProd_1974.png

InfProd_1975.png

InfProd_1976.png

InfProd_1977.png

InfProd_1978.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_1979.png

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

InfProd_1989.png

InfProd_1990.png

InfProd_1991.png

InfProd_1992.png

InfProd_1993.png

InfProd_1994.png

InfProd_1995.png

InfProd_1996.png

InfProd_1997.png

InfProd_1998.png

InfProd_1999.png

InfProd_2000.png

InfProd_2001.png

InfProd_2002.png

InfProd_2003.png

InfProd_2004.png

InfProd_2005.png

InfProd_2006.png

InfProd_2007.png

InfProd_2008.png

InfProd_2009.png

InfProd_2010.png

InfProd_2011.png

InfProd_2012.png

InfProd_2013.png

InfProd_2014.gif

InfProd_2015.png

InfProd_2016.png

InfProd_2017.gif

InfProd_2018.png

InfProd_2019.png

InfProd_2020.png

InfProd_2021.png

InfProd_2022.png

InfProd_2023.png

InfProd_2024.png

InfProd_2025.png

InfProd_2026.png

InfProd_2027.png

InfProd_2028.png

InfProd_2029.png

InfProd_2030.png

InfProd_2031.png

InfProd_2032.png

InfProd_2033.png

InfProd_2034.png

InfProd_2035.png

InfProd_2036.png

InfProd_2037.png

InfProd_2038.png

InfProd_2039.png

InfProd_2040.png

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

InfProd_2071.png

InfProd_2072.png

InfProd_2073.png

InfProd_2074.png

InfProd_2075.png

InfProd_2076.png

InfProd_2077.png

InfProd_2078.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_2079.png

InfProd_2080.png

InfProd_2081.png

InfProd_2082.gif

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

InfProd_2105.png

InfProd_2106.png

InfProd_2107.png

InfProd_2108.png

InfProd_2109.png

InfProd_2110.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_2111.png

InfProd_2112.png

InfProd_2113.png

InfProd_2114.png

InfProd_2115.png

derived from above series :

InfProd_2116.png

InfProd_2117.png

Series over prime numbers :

InfProd_2118.png

InfProd_2119.png

InfProd_2120.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_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.gif

InfProd_2142.png

InfProd_2143.png

InfProd_2144.png

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

InfProd_2145.png

InfProd_2146.png

InfProd_2147.png

InfProd_2148.png

InfProd_2149.png

InfProd_2150.png

InfProd_2151.png

InfProd_2152.png

InfProd_2153.png

InfProd_2154.png

InfProd_2155.png

InfProd_2156.png

InfProd_2157.png

InfProd_2158.png

InfProd_2159.png

InfProd_2160.png

InfProd_2161.png

InfProd_2162.png

InfProd_2163.png

InfProd_2164.png

InfProd_2165.png

InfProd_2166.png

InfProd_2167.png

InfProd_2168.png

InfProd_2169.png

InfProd_2170.png

InfProd_2171.png

InfProd_2172.png

InfProd_2173.png

InfProd_2174.png

InfProd_2175.png

InfProd_2176.png

InfProd_2177.png

InfProd_2178.png

InfProd_2179.png

InfProd_2180.png

InfProd_2181.png

InfProd_2182.png

InfProd_2183.png

InfProd_2184.png

InfProd_2185.png

InfProd_2186.png

InfProd_2187.png

InfProd_2188.png

InfProd_2189.png

InfProd_2190.png

InfProd_2191.png

InfProd_2192.png

Some ArcTan Identities :

InfProd_2193.gif

InfProd_2194.png

Series of Bessel Functions :   ( Back to Top )

InfProd_2195.png

InfProd_2196.png

InfProd_2197.png

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.png

InfProd_2232.png

InfProd_2233.png

InfProd_2234.png

InfProd_2235.png

InfProd_2236.png

InfProd_2237.png

InfProd_2238.png

InfProd_2239.png

InfProd_2240.png

InfProd_2241.png

InfProd_2242.png

InfProd_2243.png

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

InfProd_2244.png

InfProd_2245.png

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

InfProd_2246.png

InfProd_2247.png

InfProd_2248.png

InfProd_2249.png

InfProd_2250.png

InfProd_2251.png

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

InfProd_2280.png

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

InfProd_2293.png

InfProd_2294.png

InfProd_2295.png

InfProd_2296.png

InfProd_2297.png

InfProd_2298.png

InfProd_2299.gif

InfProd_2300.png

InfProd_2301.png

InfProd_2302.png

InfProd_2303.png

InfProd_2304.png

InfProd_2305.png

Series of Legendre Polynomials :   ( Back to Top )

InfProd_2306.png

InfProd_2307.png

InfProd_2308.png

InfProd_2309.png

InfProd_2310.png

InfProd_2311.png

InfProd_2312.png

InfProd_2313.png

InfProd_2314.png

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

Laguerre Polynomials:

InfProd_2329.png

Laguerre Polynomials with negative Index

InfProd_2330.png

Series of Jacobi Polynomials :

InfProd_2331.png

Series of Hermite Polynomials :

InfProd_2332.png

InfProd_2333.png

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

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

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.gif

Special Values of Zeta :

InfProd_2372.png

InfProd_2373.png

InfProd_2374.png

PolyGamma :

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.png

InfProd_2389.png

InfProd_2390.png

InfProd_2391.png

InfProd_2392.png

InfProd_2393.png

InfProd_2394.png

InfProd_2395.png

PolyLog and  LerchPhi :

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.gif

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

Special Values of PolyLog and LerchPhi :

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

InfProd_2453.png

InfProd_2454.png

InfProd_2455.png

InfProd_2456.png

InfProd_2457.png

InfProd_2458.png

InfProd_2459.png

InfProd_2460.png

InfProd_2461.png

Series of Beta Functions :   ( Back to Top )

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

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

InfProd_2490.png

InfProd_2491.png

InfProd_2492.png

InfProd_2493.png

InfProd_2494.png

InfProd_2495.png

InfProd_2496.png

InfProd_2497.png

Special values of InverseBetaRegularized :

InfProd_2498.png

Series of Gamma Functions :   ( Back to Top )

InfProd_2499.png

Dougall' s Formula :

InfProd_2500.png

InfProd_2501.png

InfProd_2502.png

InfProd_2503.png

InfProd_2504.gif

InfProd_2505.png

InfProd_2506.png

InfProd_2507.png

( K[x] = EllipticK[x], E[x] = EllipticE[x] ) :

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

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

InfProd_2520.png

InfProd_2521.png

InfProd_2522.png

InfProd_2523.png

InfProd_2524.png

InfProd_2525.png

InfProd_2526.png

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

InfProd_2527.png

InfProd_2528.png

InfProd_2529.png

InfProd_2530.png

InfProd_2531.png

InfProd_2532.png

InfProd_2533.png

Gamma Identity :

InfProd_2534.png

Special value of Gamma :

InfProd_2535.png

Special values of InverseGammaRegularized :

InfProd_2536.png

Series involving HarmonicNumber : ( Back To Top )

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

For n = 1 to 10    InfProd_2559.png is:

1: InfProd_2560.png -0.58224053
2: InfProd_2561.png -0.90797054
3: InfProd_2562.png -1.13055188
4: InfProd_2563.png -1.29927612
5: InfProd_2564.png -1.43505814
6: InfProd_2565.png -1.54863772
7: InfProd_2566.png -1.64624639
8: InfProd_2567.png -1.73181782
9: InfProd_2568.png -1.80799286
10: InfProd_2569.png -1.87662974

InfProd_2570.png

InfProd_2571.png

For n = 1 to 4     InfProd_2572.png is:

1: InfProd_2573.png 2.40411381
2: InfProd_2574.png 3.30565648
3: InfProd_2575.png 3.88459579
4: InfProd_2576.png 4.31204500

InfProd_2577.png

InfProd_2578.png

For n = 1 to 3     InfProd_2579.png is:

1: InfProd_2580.png -0.7512856
2: InfProd_2581.png -1.1496340
3: InfProd_2582.png -1.4185815

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

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_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

InfProd_2625.png

InfProd_2626.png

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

Some Limits :   ( Back to Top )

InfProd_2658.png

InfProd_2659.png

InfProd_2660.png

InfProd_2661.png

InfProd_2662.png

InfProd_2663.png

InfProd_2664.png

A few Integrals :   ( Back to Top )

InfProd_2665.png

Substitute  InfProd_2666.png   and the Feynman - Hibbs Integral

InfProd_2667.png

InfProd_2668.png

and derivatives :

InfProd_2669.png is the mth derivative with respect to a :

InfProd_2670.png

InfProd_2671.png

Iterated Expressions  ( Tetration ) :   ( Back to Top )

InfProd_2672.gif

InfProd_2673.png

InfProd_2674.gif

InfProd_2675.png

InfProd_2676.png

InfProd_2677.png

The above function f[x] = - ProductLog[-Log[x]] / Log[x] has a special 'swapping' symmetry of basis and exponent in its argument: InfProd_2678.png
f[x] is not defined beyond the maximum of its inverse function InfProd_2679.png, namely  InfProd_2680.png< x, so with this symmetry it is plausible that the exponential tower
doesn't converge for x < InfProd_2681.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_2682.png

InfProd_2683.png

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

InfProd_2684.png

These carry over with a = 0 to PolyLog :

InfProd_2685.png

InfProd_2686.png

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

InfProd_2687.png

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

InfProd_2688.png

The complex decomposition of  InfProd_2689.png with 1 ≤ x ∈ R and 0 ≤ {b, s} ∈ N into real and imaginary part can be obtained by the following expression :

InfProd_2690.png

explicitly for low s and b = 2 :

InfProd_2691.png InfProd_2692.png
InfProd_2693.png InfProd_2694.png
InfProd_2695.png InfProd_2696.png
InfProd_2697.png InfProd_2698.png
InfProd_2699.png InfProd_2700.png
InfProd_2701.png InfProd_2702.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_2703.png

The real part of  InfProd_2704.pngwith 1 ≤ x ∈ R is also given by

InfProd_2705.png

For (b ∈ N) is

InfProd_2706.png

The real and imaginary parts of LerchPhi[ InfProd_2707.png, 2, 1/2 ] (on the unit circle) are

InfProd_2708.png

With Clausen type functions for LerchPhi defined as

InfProd_2709.png

InfProd_2710.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_2711.pngInfProd_2712.png (on the unit circle) are

InfProd_2713.png

the expressions for InfProd_2714.png with lowest s being

InfProd_2715.png InfProd_2716.png
InfProd_2717.png InfProd_2718.png
InfProd_2719.png InfProd_2720.png
InfProd_2721.png InfProd_2722.png
InfProd_2723.png InfProd_2724.png
InfProd_2725.png InfProd_2726.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_2727.png

The function InfProd_2728.png has an interesting derivative :

InfProd_2729.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_2730.png

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

InfProd_2731.png

With Clausen type functions defined as

InfProd_2732.png

InfProd_2733.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_2734.png (on the unit circle) are

InfProd_2735.png

the expressions for InfProd_2736.png with lowest s being

InfProd_2737.png

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

InfProd_2738.png

As before the derivative InfProd_2739.pngInfProd_2740.pngis InfProd_2741.png with lowered index.
The first non polynomial partnerfunctions are found to be

InfProd_2742.png

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

InfProd_2743.png

explicitly for low s :

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

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

InfProd_2758.png

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

InfProd_2759.png

InfProd_2760.png

InfProd_2761.png

InfProd_2762.png

InfProd_2763.png

InfProd_2764.png

InfProd_2765.png

InfProd_2766.png

The lowest Bernoulli and Euler Polynomials are

BernoulliB EulerE
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.png
InfProd_2782.png InfProd_2783.png InfProd_2784.png

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

InfProd_2785.png

Connection to Bernoulli and Euler numbers :

InfProd_2786.png

Clausen functions and integral :

InfProd_2787.png

InfProd_2788.png

InfProd_2789.png

InfProd_2790.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 Integral