Series associated with a forgotten identity of N\"orlund

TL;DR

This paper revisits Nörlund's series expression for the psi function to derive closed-form formulas for infinite series involving inverse factorials, Catalan numbers, and harmonic numbers, including new series for pi.

Contribution

It introduces new closed-form series involving inverse factorials, Catalan numbers, harmonic numbers, and pi, based on Nörlund's series expression for the psi function.

Findings

Derived closed-form expressions for series with inverse factorials.

Established new series involving Catalan numbers and harmonic numbers.

Proved novel series representations for pi.

Abstract

We apply a seemingly forgotten series expression of N\"orlund for the psi function to express infinite series involving inverse factorials in closed form. Many of such series contain products of Catalan numbers and (odd) harmonic numbers. We also prove some new series for $\pi$.