KaydedilenlerDerginizi ekleyin

Dergiler / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 53 Issue: 4

Evaluation formulas for the Tornheim and Euler-type double series

Emre Çay(AKDENIZ UNIVERSITY)0000-0001-9352-373X
Mümün Can(AKDENIZ UNIVERSITY)0000-0002-7149-4816
Levent Kargın(AKDENIZ UNIVERSITY)
Sayfa
—
Yayın Tarihi
2024-08-27
🏛️Bu makalenin bibliyografik verileri ve tam metin bağlantısı DergiPark üzerinden sağlanmıştır.
DergiPark Kaydı ↗
✨

Yapay Zeka Hızlı Bakış & Çıkarımlar

AI Insight
⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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.

📌 3 Maddede Bu Makale

🎯
Amaç & Hipotez: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.
📊
Temel Bulgular & Veri: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.
💡
Sonuç & Katkı: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.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Tornheim series · Euler sum · zeta function

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.

🔖

Bu Makaleye Atıf Yap

Çay, E., Can, M., Kargın, L. (2024). Evaluation formulas for the Tornheim and Euler-type double series. Hacettepe Journal of Mathematics and Statistics, Volume: 53 Issue: 4. https://doi.org/10.15672/hujms.1165578

📄 Tam Metin PDF Okuyucu

📄

PDF görüntüleyici hazırlanıyor...

Kaynakça (41)

  1. V. Adamchik, On Stirling numbers and Euler sums, J. Comput. Appl. Math. 79 (1), 119–130, 1997.
  2. T. Arakawa and M. Kaneko, On multiple L-values, J. Math. Soc. Japan 56 (4), 967– 991, 2004.
  3. A. Basu, On the evaluation of Tornheim sums and allied double sums, Ramanujan J. 26, 193–207, 2011.
  4. D. Borwein, J. M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinb. Math. Soc. 38 (2), 277–294, 1995.
  5. J. M. Borwein, I. J. Zucker and J. Boersma, The evaluation of character Euler double sums, Ramanujan J. 15, 377–405, 2008.
  6. K. N. Boyadzhiev, Evaluation of Euler-Zagier sums, Int. J. Math. Math. Sci. 27 (7), 404–412, 2001.
  7. K. N. Boyadzhiev, Consecutive evaluation of Euler sums, Int. J. Math. Math. Sci. 29 (9), 555–561, 2002.
  8. M. Can, Reciprocity formulas for Hall-Wilson-Zagier type Hardy–Berndt sums, Acta Math. Hungar. 163, 118–139, 2021.
  9. M. Cenkci and A. Ünal, A two-variable Dirichlet series and its applications, Quaest. Math. 44 (12), 1661–1679, 2021.
  10. J. Choi and H. Srivastava. Explicit evaluation of Euler and related sums, Ramanujan J. 10, 51–70, 2005.
  11. A. Dil and K. N. Boyadzhiev, Euler sums of hyperharmonic numbers, J. Number Theory 147, 490–498, 2015.
  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. O. Espinosa and V. H. Moll, The evaluation of Tornheim double sums, J. Number Theory 116, 200–229, 2006.
  14. P. Flajolet and B. Salvy, Euler sums and contour integral representations, Exp. Math. 7, 15–35, 1998.
  15. J. G. Huard, K. S. Williams and Z. Nan-Yue, On Tornheim’s double series, Acta Arith. 75 (2), 105–117, 1996.
  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. M.-S. Kim, On the special values of Tornheım’s multiple series, J. Appl. Math. & Informatics 33 (3-4), 305–315, 2015.
  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. 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. L. J. Mordell, On the evaluation of some multiple series, J. London Math. Soc. 33, 368–371, 1958.
  21. T. Nakamura, A functional relation for the Tornheim double zeta function, Acta Arith. 125 (3), 257–263, 2006.
  22. T. Nakamura, Double Lerch series and their functional relations, Aequationes Math. 75 (3), 251–259, 2008.
  23. T. Nakamura, Double Lerch value relations and functional relations for Witten zeta functions, Tokyo J. Math. 31 (2), 551–574, 2008.
  24. T. Nakamura, Symmetric Tornheim double zeta functions. Abh. Math. Semin. Univ. Hambg. 91, 5–14, 2021.
  25. T. Nakamura and K. Tasaka, Remarks on double zeta values of level 2, J. Number Theory 133, 48–54, 2013.
  26. N. Nielsen, Handbuch der Theorie der Gammafunktion, Reprinted by Chelsea Publishing Company, Bronx, New York. 1965.
  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. 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. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht-Boston-London, 2001.
  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. L. Tornheim, Harmonic double series, Amer. J. Math. 72, 303–314, 1950.
  32. H. Tsumura, On some combinatorial relations for Tornheim’s double series, Acta Arith. 105, 239–252, 2002.
  33. H. Tsumura, On alternating analogues of Tornheim’s double series, Proc. Amer. Math. Soc. 131, 3633–3641, 2003.
  34. H. Tsumura, On evaluation formulas for double L-values, Bull. Aust. Math. Soc. 70 (2), 2004, 213-221.
  35. H. Tsumura, Evaluation formulas for Tornheim’s type of alternating double series, Math. Comp. 73, 251–258, 2004.
  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. H. Tsumura, On alternating analogues of Tornheim’s double series II, Ramanujan J. 18, 81–90, 2009.
  38. C. Xu and Z. Li, Tornheim type series and nonlinear Euler sums, J. Number Theory 174, 40–67, 2017.
  39. J. Yang and Y. Wang, Summation formulae in relation to Euler sums, Integral Transforms Spec. Funct. 28 (5), 336–349, 2017.
  40. W. Wang and L. Yanhong, Euler sums and Stirling sums, J. Number Theory 185, 160–193, 2018.
  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.