Evaluation formulas for the Tornheim and Euler-type double series

Authors : Emre Çay, Mümün Can, Levent Kargın

Pages : 926-941

Doi:10.15672/hujms.1165578

View : 216 | Download : 365

Publication Date : 2024-08-27

Article Type : Research Paper

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