TY - JOUR T1 - Evaluation formulas for the Tornheim and Euler-type double series AU - Can, Mümün AU - Çay, Emre AU - Kargın, Levent PY - 2024 DA - August DO - 10.15672/hujms.1165578 JF - Hacettepe Journal of Mathematics and Statistics PB - Hacettepe University WT - DergiPark SN - 2651-477X SP - 926 EP - 941 VL - 53 IS - 4 LA - en AB - 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. KW - Tornheim series KW - Euler sum KW - zeta function KW - Bernoulli polynomial KW - Fourier series CR - [1] V. 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