[Paper Review] Tornheim-like series, harmonic numbers and zeta values

Ilham A. Aliev, Ayhan Dil|arXiv (Cornell University)|Aug 6, 2020

Advanced Mathematical Identities14 references4 citations

TL;DR

This paper provides explicit evaluations of Tornheim-like double and multiple series involving harmonic numbers and zeta values, using integral representations and generating functions. It proves that the series $\sum_{n,m=1}^\infty \frac{H_{n+m+s}}{nm(n+m+s)}$ is rational for $s \geq 1$ and irrational for $s = 0$, with closed forms in terms of zeta values, and evaluates related series involving $\zeta(2)$ and $\zeta(3)$.

ABSTRACT

Explicit evaluations of the Tornheim-like double series in the form \[ \sum_{n,m=1}^\infty \frac{H_{n+m+s}}{nm\left( n+m+s ight)},\ s\in \mathbb{N\cup } \left\{ 0 ight\} \] and their extensions are given. Furthermore, series of the type \[ \sum_{m=1}^\infty \frac{2H_{2m+1}-H_{m}}{2m\left( 2m+1 ight)} \] and some other Tornheim-like multiple series are evaluated in terms of the zeta values.

Motivation & Objective

• To determine the rationality or irrationality of Tornheim-like series involving harmonic numbers and zeta values for integer parameters.
• To derive explicit closed-form evaluations of double and multiple series of the form $\sum \frac{H_{n+m+s}}{nm(n+m+s)}$ in terms of zeta functions.
• To extend known results on Euler sums and Tornheim series by incorporating generalized harmonic numbers and shift parameters.
• To evaluate special series involving $\zeta(2)$ and $\zeta(3)$ with half-integer shifts in denominators.

Proposed method

• Transforming the double series into a triple integral using integral representations of reciprocals and logarithmic functions.
• Applying the identity $\sum_{k=1}^{n+m+s} \frac{1}{k} = \sum_{k=1}^\infty \left( \frac{1}{k} - \frac{1}{k+n+m+s} \right)$ to express harmonic numbers as series.
• Using the integral formula $\int_0^1 x^{k-1} dx = \frac{1}{k}$ to convert rational terms into integrals over the unit interval.
• Employing generating functions and series expansions such as $\sum_{n=1}^\infty (xt)^{n-1} = \frac{1}{1-xt}$ for $|xt| < 1$.
• Applying the substitution $u = \frac{1+\sqrt{t}}{1-\sqrt{t}}$ to simplify logarithmic integrals involving $\ln^3(1-t)$.
• Using the identity $\int_0^1 t^{j} \ln^3 t \, dt = -\frac{6}{(j+1)^4}$ to relate integrals to zeta values.

Experimental results

Research questions

• RQ1 Is the Tornheim-like series $\sum_{n,m=1}^\infty \frac{H_{n+m+s}}{nm(n+m+s)}$ rational or irrational for $s \in \mathbb{N} \cup \{0\}$?
• RQ2 Can explicit closed-form evaluations be derived for such series in terms of Riemann zeta values?
• RQ3 What are the exact values of series involving $\zeta(2)$ and $\zeta(3)$ with half-integer shifts in the denominator?
• RQ4 How do generalized harmonic numbers and shift parameters affect the evaluation of multiple Tornheim-like series?

Key findings

• The series $A(s) = \sum_{n,m=1}^\infty \frac{H_{n+m+s}}{nm(n+m+s)}$ is rational for $s \geq 1$ and irrational for $s = 0$, with $A(0) = 6\zeta(4) = \frac{\pi^4}{16}$.
• For $s \geq 1$, $A(s) = 6 \sum_{j=0}^{s-1} (-1)^j \binom{s-1}{j} \frac{1}{(j+1)^4}$, a rational expression in terms of inverse fourth powers.
• The series $\sum_{m,n=0}^\infty \frac{1}{(2m+1)(2n+1)(2m+2n+1)(2m+2n+2)} = 16\zeta(2) - 14\zeta(3)$.
• The series $\sum_{m,n=0}^\infty \frac{1}{(2m+1)(2n+1)(2m+2n+3)} = \frac{1}{2}\zeta(2)$, derived via logarithmic integral identities.
• The triple series $\sum_{m,n=0}^\infty \frac{1}{(2m+1)(2n+1)(2m+2n+1)(2m+2n+2)(2m+2n+3)} = 24\zeta(2) - 28\zeta(3)$.
• The method successfully evaluates multiple Tornheim-like series with half-integer shifts by transforming them into integrals involving $\ln^3(1-t)$ and $\ln^2\left(\frac{1+t}{1-t}\right)$.