Evaluation formulas for the Tornheim and Euler-type double series

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

References

1. [1] V. Adamchik, On Stirling numbers and Euler sums, J. Comput. Appl. Math. 79 (1), 119–130, 1997.
2. [2] T. Arakawa and M. Kaneko, On multiple L-values, J. Math. Soc. Japan 56 (4), 967– 991, 2004.
3. [3] A. Basu, On the evaluation of Tornheim sums and allied double sums, Ramanujan J. 26, 193–207, 2011.
4. [4] D. Borwein, J. M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinb. Math. Soc. 38 (2), 277–294, 1995.
5. [5] J. M. Borwein, I. J. Zucker and J. Boersma, The evaluation of character Euler double sums, Ramanujan J. 15, 377–405, 2008.
6. [6] K. N. Boyadzhiev, Evaluation of Euler-Zagier sums, Int. J. Math. Math. Sci. 27 (7), 404–412, 2001.
7. [7] K. N. Boyadzhiev, Consecutive evaluation of Euler sums, Int. J. Math. Math. Sci. 29 (9), 555–561, 2002.
8. [8] M. Can, Reciprocity formulas for Hall-Wilson-Zagier type Hardy–Berndt sums, Acta Math. Hungar. 163, 118–139, 2021.

9. [9] M. Cenkci and A. Ünal, A two-variable Dirichlet series and its applications, Quaest. Math. 44 (12), 1661–1679, 2021.
10. [10] J. Choi and H. Srivastava. Explicit evaluation of Euler and related sums, Ramanujan J. 10, 51–70, 2005.
11. [11] A. Dil and K. N. Boyadzhiev, Euler sums of hyperharmonic numbers, J. Number Theory 147, 490–498, 2015.
12. [12] A. Dil, I. Mezo and M. Cenkci, Evaluation of Euler-like sums via Hurwitz zeta values, Turkish J. Math. 41, 1640–1655, 2017.
13. [13] O. Espinosa and V. H. Moll, The evaluation of Tornheim double sums, J. Number Theory 116, 200–229, 2006.
14. [14] P. Flajolet and B. Salvy, Euler sums and contour integral representations, Exp. Math. 7, 15–35, 1998.
15. [15] J. G. Huard, K. S. Williams and Z. Nan-Yue, On Tornheim’s double series, Acta Arith. 75 (2), 105–117, 1996.
16. [16] S.-Y. Kadota, T. Okamoto and K. Tasaka, Evaluation of Tornheim’s type of double series, Illinois J. Math. 61 (1-2), 171–186, 2017.
17. [17] M.-S. Kim, On the special values of Tornheım’s multiple series, J. Appl. Math. & Informatics 33 (3-4), 305–315, 2015.
18. [18] Z. Li, On functional relations for the alternating analogues of Tornheim’s double zeta function, Chin. Ann. Math. Ser. B 36 (6), 907–918, 2015.
19. [19] K. Matsumoto, T. Nakamura, H. Ochiai and H. Tsumura, On value-relations, functional relations and singularities of Mordell-Tornheim and related triple zetafunctions, Acta Arith. 132 (2), 99–125, 2008.
20. [20] L. J. Mordell, On the evaluation of some multiple series, J. London Math. Soc. 33, 368–371, 1958.
21. [21] T. Nakamura, A functional relation for the Tornheim double zeta function, Acta Arith. 125 (3), 257–263, 2006.
22. [22] T. Nakamura, Double Lerch series and their functional relations, Aequationes Math. 75 (3), 251–259, 2008.
23. [23] T. Nakamura, Double Lerch value relations and functional relations for Witten zeta functions, Tokyo J. Math. 31 (2), 551–574, 2008.
24. [24] T. Nakamura, Symmetric Tornheim double zeta functions. Abh. Math. Semin. Univ. Hambg. 91, 5–14, 2021.
25. [25] T. Nakamura and K. Tasaka, Remarks on double zeta values of level 2, J. Number Theory 133, 48–54, 2013.
26. [26] N. Nielsen, Handbuch der Theorie der Gammafunktion, Reprinted by Chelsea Publishing Company, Bronx, New York. 1965.
27. [27] T. Okamoto, Multiple zeta values related with the zeta-function of the root system of type $A_{2},$ $B_{2}$ and $G_{2}$, Comment. Math. Univ. St. Pauli 61 (1), 9–27, 2012.
28. [28] J. Quan, C. Xu and X. Zhang, Some evaluations of parametric Euler type sums of harmonic numbers, Integral Transforms and Special Functions 34 (2), 162–179, 2023.
29. [29] H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht-Boston-London, 2001.
30. [30] M. V. Subbarao and R. Sitaramachandrarao, On some infinite series of L. J. Mordell and their analogues, Pacific J. Math. 119, 245–255, 1985.
31. [31] L. Tornheim, Harmonic double series, Amer. J. Math. 72, 303–314, 1950.
32. [32] H. Tsumura, On some combinatorial relations for Tornheim’s double series, Acta Arith. 105, 239–252, 2002.
33. [33] H. Tsumura, On alternating analogues of Tornheim’s double series, Proc. Amer. Math. Soc. 131, 3633–3641, 2003.
34. [34] H. Tsumura, On evaluation formulas for double L-values, Bull. Aust. Math. Soc. 70 (2), 2004, 213-221.
35. [35] H. Tsumura, Evaluation formulas for Tornheim’s type of alternating double series, Math. Comp. 73, 251–258, 2004.
36. [36] H. Tsumura, On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function, Math. Proc. Cambridge Philos. Soc. 142, 395– 405, 2007.
37. [37] H. Tsumura, On alternating analogues of Tornheim’s double series II, Ramanujan J. 18, 81–90, 2009.
38. [38] C. Xu and Z. Li, Tornheim type series and nonlinear Euler sums, J. Number Theory 174, 40–67, 2017.
39. [39] J. Yang and Y. Wang, Summation formulae in relation to Euler sums, Integral Transforms Spec. Funct. 28 (5), 336–349, 2017.
40. [40] W. Wang and L. Yanhong, Euler sums and Stirling sums, J. Number Theory 185, 160–193, 2018.
41. [41] X. Zhou, T. Cai and D. Bradley, Signed q-analogs of Tornheim’s double series, Proc. Amer. Math. Soc. 136 (8), 2689–2698, 2008.