Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2020; 57(2): 489-508

Online first article August 2, 2019      Printed March 31, 2020

https://doi.org/10.4134/BKMS.b190357

Copyright © The Korean Mathematical Society.

Evaluations of some quadratic Euler sums

Xin Si, Ce Xu

Xiamen University of Technology; Xiamen University

Abstract

This paper develops an approach to the evaluation of quadratic Euler sums that involve harmonic numbers. The approach is based on simple integral computations of polylogarithms. By using the approach, we establish some relations between quadratic Euler sums and linear sums. Furthermore, we obtain some closed form representations of quadratic sums in terms of zeta values and linear sums. The given representations are new.

Keywords: Harmonic numbers, polylogarithm function, Euler sum, Riemann zeta function

MSC numbers: Primary 11M06, 11M32, 33B15

Supported by: This work was co-supported by the National Natural Science Foundation of China (No.41801219), Natural Science Foundation of Fujian Province (No.2019J01849).