Dirichlet series and series with Stirling numbers

Khristo Boyadzhiev

doi:10.56754/0719-0646.2501.103

DOI:

https://doi.org/10.56754/0719-0646.2501.103

Abstract

This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind. As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers, derangement numbers, binomial coefficients, central binomial coefficients, and Catalan numbers.

Keywords

Series identities , Stirling numbers of the first kind , harmonic numbers , hyperharmonic numbers , Cauchy numbers , derangement numbers , Catalan numbers , central binomial coefficients

Mathematics Subject Classification:

11B73 , 30B50 , 33B30 , 40A30 , 65B10

Khristo Boyadzhiev Department of Mathematics, Ohio Northern University, Ada, Ohio 45810, USA. https://orcid.org/0000-0003-1948-6699

Pages: 103–119
Date Published: 2023-04-24
Vol. 25 No. 1 (2023)

V. Adamchik, “On Stirling numbers and Euler sums”, J. Comput. Appl. Math., vol. 79, no. 1, pp. 119–130, 1997.

K. N. Boyadzhiev, “Stirling numbers and inverse factorial series”, 2020, arXiv: 2012.14546v1.

K. N. Boyadzhiev, “New series identities with Cauchy, Stirling, and harmonic numbers, and Laguerre polynomials”, J. Integer Seq., vol. 23, no. 11, Art. 20.11.7, 2020.

K. N. Boyadzhiev, “Series with central binomial coefficients, Catalan Numbers, and harmonic numbers”, J. Integer Seq., vol. 15, no. 1, Article 12.1.7, 2012.

J. M Campbell, J. D’Aurizio and J. Sondow, “Hypergeometry of the parbelos”, Amer. Math. Montly, vol. 127, no. 1, pp. 23–32, 2020.

J. M. Campbell, J. D’Aurizio and J. Sondow, “On the interplay among hypergeometric functions, complete elliptic integrals and Fourier-Legendre series expansions”, J. Math. Anal. Appl., vol. 479, no. 1, pp. 90–121, 2019.

L. Comtet, Advanced Combinatorics, Dordrecht: D. Reidel Publishing Co., 1974.

J. H. Conway and R. Guy, The Book of Numbers, New York: Copernicus, 1996.

H. W. Gould and T. Shonhiwa, “A catalog of interesting Dirichlet series”, Missouri J. Math. Sci., vol. 20, no. 1, pp. 1–17, 2008.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, New York: Addison- Wesley Publishing Company, 1994.

G. H. Hardy and M. Riesz, The general theory of Dirichlet’s series, Cambridge: Cambridge University Press, 1915.

C. Jordan, Calculus of finite differences, New York: Chelsea Publishing Co., 1950.

D. H. Lehmer, “Interesting series involving the central binomial coefficient”, Amer. Math. Monthly, vol. 92, no. 7, pp. 449–457, 1985.

D. Merlini, R. Sprugnoli and M. C. Verri, “The Cauchy numbers”, Discrete Math., vol. 306, no. 16, pp. 1906–1920, 2006.

I. Tweddle, James Stirling’s methodus differentialis, New York: Springer, 2003.

W. Wang and Y. Chen, “Explicit formulas of sums involving harmonic numbers and Stirling numbers”, J. Difference Equ. Appl., vol. 26, no. 9–10, pp. 1369–1397, 2020.

W. Wang and C. Xu, “Alternating multiple zeta values, and explicit formulas of some Euler– Apéry–type series”, European J. Combin., vol. 93, Paper No. 103283, 25 pages, 2021.