Evaluation formulas for the Tornheim and Euler-type double series

Year 2024, Volume: 53 Issue: 4, 926 - 941, 27.08.2024

Emre Çay , Mümün Can , Levent Kargın

https://doi.org/10.15672/hujms.1165578

Abstract

We give closed-form evaluation formulas for the real and imaginary parts of the series $\sum_{m,n=1}^{\infty}\frac{e^{2\pi i\left( mx-ny\right) }} {m^{p}n^{r}\left( mc+n\right) ^{q}},$ $c\in\mathbb{N},$ in terms of certain zeta values. Particular choices of $x$ and $y$ lead to evaluation formulas for some Tornheim-type $\sum_{m,n=1}^{\infty}\frac{1}{m^{p}n^{r}\left( mc+n\right) ^{q}}$ and Euler-type $\sum_{m,n=1}^{\infty}\frac{1}{n^{p}\left( mc+n\right) ^{q}}$ double series and their alternating analogues.

Keywords

Tornheim series, Euler sum, zeta function, Bernoulli polynomial, Fourier series

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