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.\textbf{ }Particular choices of $x$ and $y$ lead 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.
Tornheim series Euler sum zeta functions Bernoulli polynomial Fourier series
Birincil Dil | İngilizce |
---|---|
Konular | Matematik |
Bölüm | Matematik |
Yazarlar | |
Erken Görünüm Tarihi | 14 Eylül 2023 |
Yayımlanma Tarihi | |
Yayımlandığı Sayı | Yıl 2024 Erken Görünüm |