Evaluation formulas for the Tornheim and Euler-type double series
Hacettepe Journal of Mathematics and Statistics, vol.53, no.4, pp.926-941, 2024 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 53 Issue: 4
- Publication Date: 2024
- Doi Number: 10.15672/hujms.1165578
- Journal Name: Hacettepe Journal of Mathematics and Statistics
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Page Numbers: pp.926-941
- Keywords: Bernoulli polynomial, Euler sum, Fourier series, Tornheim series, zeta function
- Akdeniz University Affiliated: Yes
Abstract
We give closed-form evaluation formulas for the real and imaginary parts of the series ∑∞m,n=1 e2πi(mx−ny) mpnr q (mc+n), c ∈ N, in terms of certain zeta values. Particular choices of x and y lead to evaluation formulas for some Tornheim-type∑∞ 1 m,n=1 and Euler-type ∑∞m,n=1 1 np(mc+n)q double series and their alternating analogues. mpnr (mc+n)q.