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:

- Products involving Theta Functions

- q-Series

- special values of EllipticK and EllipticE

- Series of Hyperbolic Functions

- Series of Inverse Tangents ( Arcustangent )

- Series of Legendre Polynomials

- Series of Zeta PolyGamma PolyLog and related

- Series involving HarmonicNumber

- Series involving Hypergeometric Functions

- iterated expressions ( Tetration )

- some properties of ProductLog LerchPhi and PolyLog

{j, n, m} are Integer; {λ, q} > 0 and r are real; {z, , , , } may be complex; Γ[a] is Gamma[a];

sl[x] cl[x] and ϖ denote the lemniscate functions and constant ;

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 (Rn) and are referenced below at the bottom.

Infinite Products : ( Back to Top )

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:

with n = 2 :

most common case (a quadratic binomial)

with and :

Decomposition of the general quadratic trinomial applying the shorthands

gives

Order 3:

with n = 3 :

With 4 abbreviations

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

Order 4:

A simple one :

And at last using these 9 subexpressions

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

as simple function of k :

Special cases with m = 0 :

q - Product (0 < q < 1) :

Two kinds of decomposition of the same product :

With

(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

or if f[k] = then

More Products :

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

Euler’s product :

The next product approximates a Gauss function with for a large value of m:

Products of trig functions :

Products containing factors built from all primes p:

Products involving Theta Functions ( Back to Top )

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

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

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 :

EllipticThetas with imaginary argument :

With z → 0 we get

Half Lambda :

Double Lambda :

Other relations :

Square and square root of q :

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

Half Argument :

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 carried out first we get products with Tanh and Coth :

The theta functions may be expressed through each other :

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

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

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 i 2 π

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

Series expansion of InverseEllipticNomeQ :

Series expansion of EllipticNomeQ :

Specific Values :

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

products with q = :

m = InverseEllipticNomeQ[] :

special cases :

Theta Functions, specific values :

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 below under ' diverse series'.

Series involving exponentials :

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) :

Series involving :

Series of trigonometric functions :

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

Amazing identities connecting trigonometric and lemniscate functions:

q - Series : ( Back to Top )

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

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.

There is a small stumble stone in the definition of LerchPhi in the neighbourhood of a = 0: .

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

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

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

For 0.2 < q is in good approximation :

other :

QFunction Identities :

With x ∈ Reals is

Real Part | Imaginary Part | |

n=1: | 0 | |

n=2: | ||

n=3: | ||

n=4: | ||

n=5: | ||

n=6: | ||

n=7: |

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

E[m] is EllipticE[m];

Series of Hyperbolic Functions: ( Back to Top )

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

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

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

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.

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.

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 watch out, A159600 excludes a factorneeded 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 |

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 ].

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.

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 :

argument addition formulae :

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 :

One can verify the following identities 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) :

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

Series of CosIntegral: ( Back to Top )

Some Limits : ( Back to Top )

diverse Series : ( Back to Top )

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

Multiple Sums (lattice sums) :

Borwein' s formula :

Benson' s formula:

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 :

The last sum above shows only asymmetric convergence, 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 means : you get the form of function g from the derivative (integral) of function f.

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 :

Some one dimensional cases :

more series :

Sum of the inverse k-gonal numbers :

Values of the series for the first k :

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)

more 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)

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 ∈ Integer.

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.

derived from above series :

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 ∈ Integer.

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

Some ArcTan Identities :

Series of Bessel Functions : ( Back to Top )

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

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

Series of Legendre Polynomials : ( Back to Top )

Series of Jacobi Polynomials :

Series of Hermite Polynomials :

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

Special Values of Zeta :

PolyGamma :

PolyLog and LerchPhi :

Special Values of PolyLog and LerchPhi :

Series of Beta Functions : ( Back to Top )

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

Special values of InverseBetaRegularized :

Series of Gamma Functions : ( Back to Top )

Dougall' s Formula :

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

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 Identity :

Special value of Gamma :

Special values of InverseGammaRegularized :

Series involving HarmonicNumber : ( Back To Top )

For n = 1 to 10 is:

1: | -0.58224053 | |

2: | -0.90797054 | |

3: | -1.13055188 | |

4: | -1.29927612 | |

5: | -1.43505814 | |

6: | -1.54863772 | |

7: | -1.64624639 | |

8: | -1.73181782 | |

9: | -1.80799286 | |

10: | -1.87662974 |

For n = 1 to 4 is:

1: | 2.40411381 | |

2: | 3.30565648 | |

3: | 3.88459579 | |

4: | 4.31204500 |

For n = 1 to 3 is:

1: | -0.7512856 | |

2: | -1.1496340 | |

3: | -1.4185815 |

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 )

A few Integrals : ( Back to Top )

Substitute and the Feynman - Hibbs Integral

and derivatives :

Iterated Expressions ( Tetration ) : ( Back to Top )

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.

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 ] .

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 makes 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 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 :

Clausens 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