Collection of Infinite Products and Series

Dr. Andreas Dieckmann, Physikalisches Institut der Uni Bonn

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:

{j, n, m} are Integer; {λ, q} > 0 and r are real; {z, , , , } may be complex; Γ[a] is  Gamma[a];
], ] are shorthands for the Elliptic Integrals,
sl[x]  cl[x] and ϖ (CurlyPi) denote the lemniscate functions and constant ;
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.

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 :

Products of two Gammas :

Partial Fraction Decompositions :

General expression :

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

Order 2:

most common  case (a quadratic binomial), with    and  :

Decomposition of the general quadratic trinomial applying the shorthands

gives

Order 3:

With 3 abbreviations

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

Order 4:

A simple one :

Decomposition according to the general formula above :

And at last using these 9  subexpressions

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

as simple function of k :

Decomposition of  finite products into power series

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) :

Special cases with m = 0 :

q - Product (0 < q < 1) :

Two kinds of decomposition of the same product :

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

(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] :

If f[k] =   then (0 < x)

or if f[k] =   then

More Products :

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

Euler’s product :

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

For a large value of m >> n the next product approximates a Gauss function with standard deviation :

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

Products of trig functions :

Vieta' s product (set x → π/2 in the first product of Cos above)  was maybe the first (anno 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 -polygon (built from isosceles triangles) inscribed into the same circle
,
beginning with n = 2. It ends up at n = ∞, where the area of the polygon is equal to the area of the circle with
.

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

and may then be compared to its lemniscatic cousin, where the multiplications inside the square roots are changed to divisions:

Products containing factors built from all primes p:

Special  values :

Special  rational  values :

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

Series and Product Representations :

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

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

Approximation of quotients (0.4 < λ) :

Partial differential equation (pde) :

EllipticThetas with imaginary argument :

With z → 0 this reduces to

Half Lambda :

Double Lambda :

Other relations :

From an equation involving Eisenstein series and and their connection to theta functions:

Square and square root of q :

Double Argument  (Landen), -  see above for double  )  and half  Lambda - :

Half Argument :

Proportionality of imaginary and real part of ,   and   with real argument  and imaginary nome:

Derivatives with respect to q :

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

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

Series of theta functions :

Now may be extracted out of the sum because of its periodicity (see table below)

{0.554084,-0.554084,0.554084,-0.554084,0.554084,-0.554084,0.554084,-0.554084,0.554084,-0.554084,0.554084}

and the remaining sum can be done :

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

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) :

Series representation of ratios of theta functions :

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

Double product representation of the single theta functions :

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

The theta functions may be expressed through each other :

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

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

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

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

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:

cf.  Hyde: A Wallis product on clovers.

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

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

and  (*1*)/(*2*) now running from k = 0  gives :

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

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

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

Trigonometric and hyperbolic Products :

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

q - Products :

In the following is ( 0 < q < 1 ) and [ 0 , q ] ,   ([ 0 , q ] =[ 0 , - q ] ) :

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

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

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

and can be expressed through m[q] , K[m[q]] and E[m[q]] :

and similarly :

and :

and from combining the above like :

we get :

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

or turned the other way round :

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 :

Series expansion of InverseEllipticNomeQ :

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

Connection of InverseEllipticNomeQ to ModularLambda :

(n) in Wikipedia (see ' modular lambda function') .

Special values of (n) :

Specific Values :

A special (remarkable) relation :

EllipticNomeQ :

Series expansion and approximation :

Square and square root of the nomen :

Specific Values :

q[m[#]] = #; from specific values of InverseEllipticNomeQ above like for example q[ m[] ]  =  q[ ]  =;

Ramanujans g functions:

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

products with q = :

m = InverseEllipticNomeQ[],   = [0, ] :

special cases :

Theta Functions, specific values :

Beauty meets well-tempered music…;-) ↓

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: means .

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

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

Series involving exponentials :

Theta functions as series of shifted Gauss functions having equal widths (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) :

Series involving :

For low values of m and n this gives (with  PL = as shorthand) :

 PL;

Series of Stirling numbers :

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

Series of trigonometric functions :

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

Sin[π a] = Cos[π (a - 1/2)] :

Amazing identities connecting trigonometric and lemniscate functions:

Special values of trigonometric functions :

Euler :

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

 1 1 1 1

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

 0 0 0 0

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

 1
 -1
 1
 -1
 1
 -1

 0
 0
 0
 0
 0
 0

Repeated angular bisection of any angle φ  :

If Cos[φ] allows a radical expression (see below), then the Cos or Sin of the repeated bisection   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)):

 sin[0] 0
 sin[0] 0
 sin[0] 0
 sin[0] 0

 cos[0] 1
 cos[0] 1
 cos[0] 1
 cos[0] 1

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

or in steps of π/10 :

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) :

 ComplexInfinity 1 ComplexInfinity -1 ComplexInfinity 1 ComplexInfinity -1

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

 0 1 0 -1 0 1 0 -1

Some (special) special values :

(0 < q < 1); With → Cosh[ k Log[ q ]] + Sinh[ k Log[ q ]] the following expressions can be transformed into sums of hyperbolic functions.

 1 1 1 1 4 1 1 11 11 1 1 26 66 26 1 1 57 302 302 57 1

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

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

Using an identity from (R4) some series involving the Floor function can be solved  ( [ 0 , q ] is an EllipticTheta function and 0 < q < 1) :

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

Lambert Type q Series:

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

The next q - series (q → ) are connected to the Eisenstein Series like

With q = this kind of sum is

For 0.2 < q is in good approximation :

other :

QFunction Identities :

Special values of QPolyGamma :

With x ∈ Reals is

 Real Part Imaginary Part n=1: 0 n=2: n=3: n=4: n=5: n=6: n=7:

K[m] is EllipticK[m];

E[m] is EllipticE[m];

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

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

Some Jacobi elliptic functions :

Special values :

Hyperbolic series involving the lemniscate functions :

ϖ is the lemniscate constant :

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:

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

 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 beware,  A159600 excludes a factor needed to obtain the series for cl) :

 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 :

Ramanujan's Cos/Cosh identity :

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):

The 'four horsemen of the apocalypse':

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?

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 ].

The QPolyGamma 'monsters' :

m = InverseEllipticNomeQ[] :

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 :

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

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

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

m = InverseEllipticNomeQ[] :

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.

m = InverseEllipticNomeQ[] :

Some hyperbolic Identities :

Some Lemniscate Sine and Cosine Identities including derivative and integral:

Periods :

Dual sibling of the Pythagorean Identity ) :

Special values :

Imaginary, negative and double arguments :

Squares :

Derivatives and basic integrals :

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] = (sl[x + dx] - sl[x])/dx = cl[x] (1 + ) 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 :

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

Complex properties :

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

The ‘lemniscatic tangent’ is then represented by :

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

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

Higher multifactorials :

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

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

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

where the recursions

and

give the integers and , 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):

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

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

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

 1 1 1 1 z z z -z

A completely crazy series :

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

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

and appears in a special combinatorics problem :
It calculates the count of possible ways 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}]

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 :

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.

Higher powers of the factorial in the denominator :

Replace with and   with
as well as   with 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 () 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 {}):

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 :

0 < j < m:

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 into sums of powers less than :

and the recursion for the coefficients c is given by :

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

For example with n = 6, m = 3 :

The s are then :

 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

so that the recursionally defined sum yields :

while Mathematica gives:

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 :

The  highest exponent of 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) 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:

 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 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]} }:

 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 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 :

where two recursions are needed:

The series

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

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

Factorial :

Borwein' s formula :

Benson' s formula:

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

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.

Double series involving lemniscate sine and cosine :

Both 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.

Some values of Eisenstein series :

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

or directly by :

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

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

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.

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.

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

Double Series involving the Zeta function :

Multiple Series involving the Zeta function :

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 :

 Zeta[3] Zeta[5]

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]) :

 Zeta[3] Zeta[3] Zeta[5] Zeta[5] Zeta[7]

The notation means that the divergent term 1/0 is excluded :

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]

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

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]

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] Log[2]

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  ) is determined by Binomial[k - 1, m - 1]:

 -Log[2]

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

and

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

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

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

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

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

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

So (for example) the lattice version of

Lattice q - sums :

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

where the summands characterized by   = k occur with a certain multiplicity given by

This tells the number of ways to express k as a sum of m integers (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

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

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

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

For m = n + 2 the sum reads :

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

 2+k 2+k

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 like:

Shown above are the 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.

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

( Zeta[n - 1] - Zeta[n] = Zeta[n - 1, 0] - Zeta[n, 0] = Zeta[n - 1, 1] - Zeta[n, 1] = Zeta[n - 1, 2] - Zeta[n, 2],   Zeta[n,1] = Zeta[n,0] = Zeta[n] )

Sum of the inverse m - gonal numbers :

Values of the series for the first m :

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

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

k - nomial triangles:

Generate the  k - nomial triangle  as coefficents of :
(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)

other series:

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

This sum alternates between ± π  for z ∈ N :

In the following 4 expressions b = :

The next three expressions contain s = and t = :

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

The sum of  integer powers  of the inverse trigonal numbers :

(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.

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

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.

Lattice Sums :

Exchange x ⇔ y in previous expression :

derived from above series :

The double sum below leads to Green' s function for the Laplace Operator in two dimensions inside a rectangle with sides a and b, the point source being located at xq, yq :

Series over prime numbers :

(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.

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

Lattice Sums :

Some ArcTan Identities :

Special values :

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

Cases of Neumann' s addition theorem :

Cases of Graf' s addition theorem :

Laguerre Polynomials:

Laguerre Polynomials with negative Index

Series of Jacobi Polynomials :

Series of Hermite Polynomials :

The next expression is an asymptotic approximation in s (better than 1 % ):

Special Values of Zeta :

PolyGamma :

PolyLog and  LerchPhi :

The sum inside the large brackets above gives the Eulerian numbers .

From Reynolds’ LerchPhi equation (4.1) in https://arxiv.org/pdf/2306.12565.pdf :

Special Values of PolyLog and LerchPhi :

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

Special  values  of  Beta related functions :

Dougall' s Formula :

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

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

Gamma Identities :

Special value of Gamma :

Special values of InverseGammaRegularized :

For n = 1 to 10     is:

 1: -0.582241 2: -0.907971 3: -1.13055 4: -1.29928 5: -1.43506 6: -1.54864 7: -1.64625 8: -1.73182 9: -1.80799 10: -1.87663

For n = 1 to 4      is:

 1: 2.40411 2: 3.30566 3: 3.8846 4: 4.31205

For n = 1 to 3      is:

 1: -0.751286 2: -1.14963 3: -1.41858

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.

Special value:

Substitute     and the Feynman - Hibbs Integral

and derivatives :

is the mth derivative with respect to a :

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

Solve special recursions using a corresponding differential equation:

Consider recursions of the form f[k+1] = f[0] g[ f[k] ] + f[k] that converge to finite values for large k. They may sometimes be solved by using a differential equation.
If  f[0] ~ x/n, where some large k is  k ≤ n, then f[0] will be small and the differential equation to try to solve reads  y'[x] == g[ y[x] ] with boundary condition y[0] = x/n.
The recursion f[n] calculated starting from a certain value of x/n will then converge to y[x].
Several cases of g[ f[k] ] are presented.  All result in strictly monotonically increasing y[x] over their respective domain.
Verify that your selection of {a,b,c} and size of n is working out by numerical and graphical check,
(The value of n may have to be inceased in some cases for better convergence; if the limit n→ ∞ leads to y = 0, keep n at a large but finite value as shown in the Sin[] and Tan[] example below):

The points of the recursion are iteratively calculated and can be subsequently plotted together with the result in this case like

You may have to adapt the Plot ranges and intervalls of x to the next exemplary cases :

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 ] .

 1 1 2 1 8 6 1 22 58 24

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

These carry over with a = 0 to PolyLog :

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

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

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

explicitly for low s and b = 2 :

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) :

The real part of  with 1 ≤ x ∈ R is also given by

For (b ∈ N) is

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

With Clausen type functions for LerchPhi defined as

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

the expressions for with lowest s being

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

The function has an interesting derivative :

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 :

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

With Clausen type functions defined as

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

the expressions for with lowest s being

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

As before the derivative is with lowered index.
The first non polynomial partnerfunctions are found to be

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

explicitly for low s :

For real x < 1  is :

 Log[x]-Log[1-x]-i π

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

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

The lowest Bernoulli and Euler Polynomials are

 BernoulliB EulerE

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

Connection to Bernoulli and Euler numbers :

Clausen functions and integral :

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

 -Log[1-x] PolyLog[2,x] PolyLog[3,x] PolyLog[4,x] PolyLog[5,x]

Jacobi:

λ→0

Sinh:

Sin:
is from the number of n negative factors in the denominator…

Cosh:

Cos:

Tan:

Tanh:

z→-z

Re[z]<>k &&Im[z]<>k

Reihenentwicklung von m:

Glaisher:

-(z→-z)

Proof double Product