Applications of Euler Sums and Series Involving the Zeta Functions

Author:

Choi Junesang1ORCID,Sofo Anthony2ORCID

Affiliation:

1. Department of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea

2. College of Engineering and Science, Victoria University, P.O. Box 14428, Footscray, VIC 3011, Australia

Abstract

A very recent article delved into and expanded the four parametric linear Euler sums, revealing that two well-established subjects—Euler sums and series involving the zeta functions—display particular correlations. In this study, we present several closed forms of series involving zeta functions by using formulas for series associated with the zeta functions detailed in the aforementioned paper. Another closed form of series involving Riemann zeta functions is provided by utilizing a known identity for a series of rational functions in the series index, expressed in terms of Gamma functions. Furthermore, we demonstrate a myriad of applications and relationships of series involving the zeta functions and the extended parametric linear Euler sums. These include connections with Wallis’s infinite product formula for π, Mathieu series, Mellin transforms, determinants of Laplacians, certain integrals expressed in terms of Euler sums, representations and evaluations of some integrals, and certain parametric Euler sum identities. The use of Mathematica for various approximation values and certain integral formulas is elaborated upon. Symmetry naturally occurs in Euler sums.

Publisher

MDPI AG

Subject

Physics and Astronomy (miscellaneous),General Mathematics,Chemistry (miscellaneous),Computer Science (miscellaneous)

Reference104 articles.

1. Berndt, B.C. (1985). Ramanujan’s Notebooks, Springer. Part I.

2. On some infinite series;Briggs;Scr. Math.,1955

3. Srivastava, H.M., and Choi, J. (2012). Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier.

4. On an intriguing integral and some series related to ζ(4);Borwein;Proc. Am. Math. Soc.,1995

5. A new method for investigating Euler sums;Basu;Ramanujan J.,2000

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