Euler Sums and Integral Connections

Abstract

In this paper, we present some Euler-like sums involving partial sums of the harmonic and odd harmonic series. First, we give a brief historical account of Euler’s work on the subject followed by notations used in the body of the paper. After discussing some alternating Euler sums, we investigate the connection of integrals of inverse trigonometric and hyperbolic type functions to generate many new Euler sum identities. We also give some new identities for Catalan’s constant, Apery’s constant and a fast converging identity for the famous ζ ( 2 ) constant.

Keywords

• Euler sums
• Catalan’s constant
• Trigamma function
• integral representation
• closed form
• ArcTan and ArcTanh functions
• partial fractions