Generalized Harmonic Number Sums and Symmetric Functions

Kwang-Wu Chen

Department of Mathematics

University of Taipei

kwchen@utaipei.edu.tw

    We express some general type of infinite series such as
`\sum_{n=1}^\infty\frac{F(H_n^{(1)}(z),H_n^{(2)}(z),\dots,H_n^{(ℓ)}(z))}{(n+z)^{s_1}(n+1+z)^{s_2}\cdots(n+k-1+z)^{s_k}}`,
where `F(x_1,\dots,x_ℓ)\in\mathbb{Q}[x_1,\dots,x_ℓ]`, `H_n^{(m)}(z)=\sum_{j=1}^n1/(j+z)^m`, and `s_1,\dots,s_k` are nonnegative integers with `s_1+\cdots+s_k\geq 2`, as a linear combination of multiple Hurwitz zeta functions and harmonic functions.

Keyword: Symmetric functions, multiple Hurwitz zeta functions

References

[1] S. Akiyama, H. Ishikawa, 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, 2002, pp. 1–16. DOI: 10.1007/978-1-4757-3621-2_1.
[2] K.-W. Chen, Generalized Harmonic Numbers and Euler Sums, Int. J. Number Theory, 13 (2) (2017), 513–528. DOI:10.1142/S1793042116500883.
[3] K.-W. Chen, C.-L. Chung, M. Eie, Sum formulas of multiple zeta values with arguments multiples of a common positive integer, J. Number Theory, 177 (2016), 479–496.
[4] M.-A. Coppo, B. Candelpergher, The Arakawa-Kaneko zeta function, Ramanujan J., 22.2 (2010), 153–162.
[5] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, second edition, Addison-Wesley, 1994.
[6] M. E. Hoffman, On multiple zeta values of even arguments, arXiv: 1205.7051v4 (2016).
[7] M. E. Hoffman, Harmonic-number summation identities, symmetric functions, and multiple zeta values, Ramanujan J., 42 (2) (2017), 501–526.
[8] J. P. Kelliher, R. Masri, Analytic continuation of multiple Hurwitz zeta functions, Math. Proc. Camb. Phil. Soc., 145 (2008), 605–617.
[9] I. G. MacDonald, Symmetric Functions and Hall Polynomials, 2nd edition, Claredon Press, 1995.
[10] J. Mehta, G. K. Viswanadham, Analytic continuation of multiple Hurwitz zeta functions, J. Math. Soc. Janpan, 69 (4) (2017), 1431–1442.
[11] A. Sofo, Harmonic number sums in higher powers, J. Math. Anal., 2 (2) (2011), 15–22.
[12] A. Sofo, M. Hassani, Quadratic harmonic number sums, Appl. Math. E-Notes, 12 (2012), 110–117.
[13] R. P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, 1999.
[14] J. Zhao, Sum formula of multiple Hurwitz-zeta values, Forum Mathematicum, 27 (2) (2015), 929-936.