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Series associated with a forgotten identity of Nörlund

Kunle Adegoke, Robert Frontczak

Abstract

We apply a seemingly forgotten series expression of Nörlund 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 $π$.

Series associated with a forgotten identity of Nörlund

Abstract

We apply a seemingly forgotten series expression of Nörlund 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 .

Paper Structure

This paper contains 3 sections, 17 theorems, 80 equations.

Table of Contents

  1. Motivation and Preliminaries
  2. First new series associated with \ref{['psi_expr3']}
  3. More new series associated with \ref{['psi_expr3']}

Key Result

Theorem 1

For each integer $k\geq 2$ we have the expression In particular, we have the curious expressions for $\pi$ in the form and

Theorems & Definitions (34)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • Lemma 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • ...and 24 more