Research Article
Year 2024,
Early Access, 1 - 16
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.\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.
References
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- T. Arakawa and M. Kaneko, On multiple L-values, J. Math. Soc. Japan 56 (4), 967--991, 2004.
- A. Basu, On the evaluation of Tornheim sums and allied double sums, Ramanujan J. 26, 193--207, 2011.
- D. Borwein, J. M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinb. Math. Soc. 38 (2), 277--294, 1995.
- J. M. Borwein, I. J. Zucker and J. Boersma, The evaluation of character Euler double sums, Ramanujan J. 15, 377--405, 2008.
- K. N. Boyadzhiev, Evaluation of Euler-Zagier sums, Int. J. Math. Math. Sci. 27 (7), 404--412, 2001.
- K. N. Boyadzhiev, Consecutive evaluation of Euler sums, Int. J. Math. Math. Sci. 29 (9), 555--561, 2002. M. Can, Reciprocity formulas for Hall-Wilson-Zagier type Hardy--Berndt sums, Acta Math. Hungar. 163, 118--139, 2021. M. Cenkci and A. Ünal, A two-variable Dirichlet series and its applications, Quaest. Math. 44 (12), 1661--1679, 2021.
- J. Choi and H. Srivastava. Explicit evaluation of Euler and related sums, Ramanujan J. 10, 51--70, 2005. A. Dil and K. N. Boyadzhiev, Euler sums of hyperharmonic numbers, J. Number Theory 147, 490--498, 2015. A. Dil, I. Mezo and M. Cenkci, Evaluation of Euler-like sums via Hurwitz zeta values, Turkish J. Math. 41, 1640--1655, 2017.
- O. Espinosa and V. H. Moll, The evaluation of Tornheim double sums, J. Number Theory 116, 200--229, 2006.
- P. Flajolet and B. Salvy, Euler sums and contour integral representations, Exp. Math. 7, 15--35, 1998.
- J. G. Huard, K. S. Williams and Z. Nan-Yue, On Tornheim's double series, Acta Arith. 75 (2), 105--117, 1996.
- 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.
- M.-S. Kim, On the special values of Tornheım's multiple series, J. Appl. Math. & Informatics 33 (3-4), 305--315, 2015.
- 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.
- K. Matsumoto, T. Nakamura, H. Ochiai and H. Tsumura, On value-relations, functional relations and singularities of Mordell-Tornheim and related triple zeta-functions, Acta Arith. 132 (2), 99--125, 2008.
- L. J. Mordell, On the evaluation of some multiple series, J. London Math. Soc. 33, 368--371, 1958.
- T. Nakamura, A functional relation for the Tornheim double zeta function, Acta Arith. 125 (3), 257--263, 2006.
- T. Nakamura, Double Lerch series and their functional relations, Aequationes Math. 75 (3), 251--259, 2008.
- T. Nakamura, Double Lerch value relations and functional relations for Witten zeta functions, Tokyo J. Math. 31 (2), 551--574, 2008.
- T. Nakamura, Symmetric Tornheim double zeta functions. Abh. Math. Semin. Univ. Hambg. 91, 5--14, 2021.
- T. Nakamura and K. Tasaka, Remarks on double zeta values of level 2, J. Number Theory 133, 48--54, 2013.
- N. Nielsen, Handbuch der Theorie der Gammafunktion, Reprinted by Chelsea Publishing Company, Bronx, New York. 1965.
- T. Okamoto, Multiple zeta values related with the zeta-function of the root system of type A₂, B₂ and G₂, Comment. Math. Univ. St. Pauli 61 (1), 9--27, 2012.
- J. Quan, C. Xu and X. Zhang, Some evaluations of parametric Euler type sums of harmonic numbers, Integral Transforms and Special Functions 2022. https://doi.org/10.1080/10652469.2022.2097671 H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht-Boston-London, 2001.
- M. V. Subbarao and R. Sitaramachandrarao, On some infinite series of L. J. Mordell and their analogues, Pacific J. Math. 119, 245--255, 1985.
- L. Tornheim, Harmonic double series, Amer. J. Math. 72, 303--314, 1950.
- H. Tsumura, On some combinatorial relations for Tornheim's double series, Acta Arith. 105, 239--252, 2002.
- H. Tsumura, On alternating analogues of Tornheim's double series, Proc. Amer. Math. Soc. 131, 3633--3641, 2003.
- H. Tsumura, On evaluation formulas for double L-values, Bull. Aust. Math. Soc. 70 (2), 2004, 213-221.
- H. Tsumura, Evaluation formulas for Tornheim's type of alternating double series, Math. Comp. 73, 251--258, 2004.
- 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.
- H. Tsumura, On alternating analogues of Tornheim's double series II, Ramanujan J. 18, 81--90, 2009.
- C. Xu and Z. Li, Tornheim type series and nonlinear Euler sums, J. Number Theory 174, 40--67, 2017.
- J. Yang and Y. Wang, Summation formulae in relation to Euler sums, Integral Transforms Spec. Funct. 28 (5), 336--349, 2017.
- W. Wang and L. Yanhong, Euler sums and Stirling sums, J. Number Theory 185, 160--193, 2018.
- X. Zhou, T. Cai and D. Bradley, Signed q-analogs of Tornheim's double series, Proc. Amer. Math. Soc. 136 (8), 2689--2698, 2008.
Year 2024,
Early Access, 1 - 16
Abstract
References
- V. Adamchik, On Stirling numbers and Euler sums, J. Comput. Appl. Math. 79 (1), 119--130, 1997.
- T. Arakawa and M. Kaneko, On multiple L-values, J. Math. Soc. Japan 56 (4), 967--991, 2004.
- A. Basu, On the evaluation of Tornheim sums and allied double sums, Ramanujan J. 26, 193--207, 2011.
- D. Borwein, J. M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinb. Math. Soc. 38 (2), 277--294, 1995.
- J. M. Borwein, I. J. Zucker and J. Boersma, The evaluation of character Euler double sums, Ramanujan J. 15, 377--405, 2008.
- K. N. Boyadzhiev, Evaluation of Euler-Zagier sums, Int. J. Math. Math. Sci. 27 (7), 404--412, 2001.
- K. N. Boyadzhiev, Consecutive evaluation of Euler sums, Int. J. Math. Math. Sci. 29 (9), 555--561, 2002. M. Can, Reciprocity formulas for Hall-Wilson-Zagier type Hardy--Berndt sums, Acta Math. Hungar. 163, 118--139, 2021. M. Cenkci and A. Ünal, A two-variable Dirichlet series and its applications, Quaest. Math. 44 (12), 1661--1679, 2021.
- J. Choi and H. Srivastava. Explicit evaluation of Euler and related sums, Ramanujan J. 10, 51--70, 2005. A. Dil and K. N. Boyadzhiev, Euler sums of hyperharmonic numbers, J. Number Theory 147, 490--498, 2015. A. Dil, I. Mezo and M. Cenkci, Evaluation of Euler-like sums via Hurwitz zeta values, Turkish J. Math. 41, 1640--1655, 2017.
- O. Espinosa and V. H. Moll, The evaluation of Tornheim double sums, J. Number Theory 116, 200--229, 2006.
- P. Flajolet and B. Salvy, Euler sums and contour integral representations, Exp. Math. 7, 15--35, 1998.
- J. G. Huard, K. S. Williams and Z. Nan-Yue, On Tornheim's double series, Acta Arith. 75 (2), 105--117, 1996.
- 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.
- M.-S. Kim, On the special values of Tornheım's multiple series, J. Appl. Math. & Informatics 33 (3-4), 305--315, 2015.
- 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.
- K. Matsumoto, T. Nakamura, H. Ochiai and H. Tsumura, On value-relations, functional relations and singularities of Mordell-Tornheim and related triple zeta-functions, Acta Arith. 132 (2), 99--125, 2008.
- L. J. Mordell, On the evaluation of some multiple series, J. London Math. Soc. 33, 368--371, 1958.
- T. Nakamura, A functional relation for the Tornheim double zeta function, Acta Arith. 125 (3), 257--263, 2006.
- T. Nakamura, Double Lerch series and their functional relations, Aequationes Math. 75 (3), 251--259, 2008.
- T. Nakamura, Double Lerch value relations and functional relations for Witten zeta functions, Tokyo J. Math. 31 (2), 551--574, 2008.
- T. Nakamura, Symmetric Tornheim double zeta functions. Abh. Math. Semin. Univ. Hambg. 91, 5--14, 2021.
- T. Nakamura and K. Tasaka, Remarks on double zeta values of level 2, J. Number Theory 133, 48--54, 2013.
- N. Nielsen, Handbuch der Theorie der Gammafunktion, Reprinted by Chelsea Publishing Company, Bronx, New York. 1965.
- T. Okamoto, Multiple zeta values related with the zeta-function of the root system of type A₂, B₂ and G₂, Comment. Math. Univ. St. Pauli 61 (1), 9--27, 2012.
- J. Quan, C. Xu and X. Zhang, Some evaluations of parametric Euler type sums of harmonic numbers, Integral Transforms and Special Functions 2022. https://doi.org/10.1080/10652469.2022.2097671 H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht-Boston-London, 2001.
- M. V. Subbarao and R. Sitaramachandrarao, On some infinite series of L. J. Mordell and their analogues, Pacific J. Math. 119, 245--255, 1985.
- L. Tornheim, Harmonic double series, Amer. J. Math. 72, 303--314, 1950.
- H. Tsumura, On some combinatorial relations for Tornheim's double series, Acta Arith. 105, 239--252, 2002.
- H. Tsumura, On alternating analogues of Tornheim's double series, Proc. Amer. Math. Soc. 131, 3633--3641, 2003.
- H. Tsumura, On evaluation formulas for double L-values, Bull. Aust. Math. Soc. 70 (2), 2004, 213-221.
- H. Tsumura, Evaluation formulas for Tornheim's type of alternating double series, Math. Comp. 73, 251--258, 2004.
- 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.
- H. Tsumura, On alternating analogues of Tornheim's double series II, Ramanujan J. 18, 81--90, 2009.
- C. Xu and Z. Li, Tornheim type series and nonlinear Euler sums, J. Number Theory 174, 40--67, 2017.
- J. Yang and Y. Wang, Summation formulae in relation to Euler sums, Integral Transforms Spec. Funct. 28 (5), 336--349, 2017.
- W. Wang and L. Yanhong, Euler sums and Stirling sums, J. Number Theory 185, 160--193, 2018.
- X. Zhou, T. Cai and D. Bradley, Signed q-analogs of Tornheim's double series, Proc. Amer. Math. Soc. 136 (8), 2689--2698, 2008.
There are 36 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | September 14, 2023 |
| Publication Date | |
| Published in Issue | Year 2024 Early Access |