Infinite series containing generalized harmonic functions

Kwang-Wu Chen and Yi-Hsuan Chen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 2, Pages 85–104
DOI: 10.7546/nntdm.2020.26.2.85-104
Full paper (PDF, 287 Kb)

Details

Authors and affiliations

Kwang-Wu Chen
Department of Mathematics, University of Taipei
No. 1, Ai-Guo West Road, Taipei 10048, Taiwan

Yi-Hsuan Chen
Department of Mathematics, University of Taipei
No. 1, Ai-Guo West Road, Taipei 10048, Taiwan

Abstract

We use Abel’s summation formula and the method of partial fraction decomposition to study infinite series involving generalized harmonic numbers of any positive integral order, with any positive integral power.

Keywords

• Abel’s summation formula
• Generalized harmonic functions
• Multiple Hurwitz zeta functions
• Digamma function

2010 Mathematics Subject Classification

• 05A19
• 11M06
• 40A25

References

1. Ablinger, J., Blümlein, J., & Schneider, C. (2011). Harmonic sums and polylogarithms generated by cyclotomic polynomials, J. Math. Phys., 52, 102301.
2. Ablinger, J., Blümlein, J., & Schneider, C. (2014). Generalized harmonic, cyclotomic, and binomial sums, their polylogarithms and special numbers, J. Phys: Conference Series, 523, 012060.
3. Akiyama, S., & Ishikawa, H. (2002). On analytic continuation of multiple L-functions and related zeta-functions, Analytic Number Theory, C. Jia and K. Matsumoto (eds.), Developments in Math. Vol. 6, Kluwer, 1–16.
4. Batir, N., & Chen, K.-W. (2019). Finite Hurwitz–Lerch Functions, Filomat, 33(1), 101–109.
5. Chen, K.-W. (2018). Generalized harmonic number sums and quasi-symmetric functions, arXiv:1812.06685 [math.NT].
6. Chen, K.-W. (2019). Delannoy numbers and preferential arrangements, Mathematics, 7 (3), 238.
7. Choi, J. (2013). Finite summation formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers, J. Inequal. Appl., 49, 11 pages.
8. Chu, W. (2012). Infinite series identities on harmonic numbers, Results Math., 61, 209–221.
9. Coffey, M.W. (2008). On a three-dimensional symmetric Ising tetrahedron, and contributions to the theory of the dilogarithm and Clausen functions, J. Math. Phys., 49, 043510-1–043510-32.
10. Davydychev, A.I., & Kalmykov, M. Yu. (2004). Massive Feynman diagrams and inverse binomial sums, Nuclear Phys. B, 699 (1–2), 3–64.
11. Hoffman, M. E. (2017). Harmonic-number summation identities, symmetric functions, and multiple zeta values, Ramanujan J., 42 (2), 501–526.
12. Hoffman, M. E. (2020). References on multiple zeta values and Euler sums, 1 April, 2020, Available online at: https://www.usna.edu/Users/math/meh/biblio.html.
13. Kalmykov, M. Y. & Veretin, O. (2000). Single-scale diagrams and multiple binomial sums, Phys. Lett. B, 483 (1–3), 315–323.
14. Kalmykov, M. Y., Ward, B. F. L., & Yost, S. A. (2007). Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter, J. High Energy Phys., 10, 048.
15. Kelliher, J. P., & Masri, R. (2008). Analytic continuation of multiple Hurwitz zeta functions, Math. Proc. Camb. Phil. Soc., 145, 605–617.
16. MacDonald, I. G. (1995). Symmetric Functions and Hall Polynomials, 2nd edition, Claredon Press.
17. Mehta, J., & Viswanadham, G. K. (2017). Analytic continuation of multiple Hurwitz zeta functions, J. Math. Soc. Japan, 69 (4), 1431–1442.
18. Sofo, A. (2009). Harmonic numbers and double binomial coefficients, Integral Transforms Spec. Funct., 20: 11, 847–857.
19. Sofo, A. (2010). Harmonic sums and integral representations, J. Appl. Analysis, 16, 265–277.
20. Sofo, A. (2011). Harmonic number sums in higher powers, J. Math. Anal., 2 (2), 15–22.
21. Sofo, A., & Hassani, M. (2012). Quadratic harmonic number sums, Applied Mathematics E-Notes, 12, 110–117.
22. Stanley, R. P. (1999). Enumerative Combinatorics. Vol. 2, Cambridge University Press.
23. Wade,W. R. (2014) Introduction to Analysis, Pearson Education Limited, 4th edition.
24. Wang, X. (2018). Infinite series containing generalized harmonic numbers, Results Math., 73, 24.
25. Wang, X., & Chu, W. (2018). Infinite series identities involving quadratic and cubic harmonic numbers, Publ. Mat., 62, 285–300.
26. Weinzierl, S. (2004). Expansion around half-integer values, binomial sums, and inverse binomial sums, J.Math. Phys., 45 (7), 2656–2673.
27. Xu, C., Zhang, M., & Zhu, W. (2016). Some evaluation of harmonic number sums, Integral Transforms Spec. Funct., 27 (12) , 937–955.