Values and recurrence relations for integrals of powers of arctan and logarithm and associated Euler-like sums

TL;DR

This paper evaluates integrals involving arctan and logarithm functions, introduces new summation identities for harmonic numbers and constants, and expresses these sums in terms of well-known mathematical constants.

Contribution

It provides novel evaluations of integrals and summation identities involving special functions and constants, expanding the understanding of these mathematical relationships.

Findings

New integral evaluations involving arctan and logarithm functions.

Summation identities expressed as finite sums of special constants.

Illustrative examples demonstrating the theorems.

Abstract

In this paper, we give evaluations of integrals involving the arctan and the logarithm functions, and present several new summation identities for odd harmonic numbers and Milgram constants. These summation identities can be expressed as finite sums of special constants such as $\pi$, the Catalan constant, the values of Riemann zeta function at the positive odd numbers and $\ln2$ etc.. Some examples are detailed to illustrate the theorems.