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On Product Formulas of Guillera and Sondow

Shihan Kanungo, Jordan Schettler

Abstract

In this note, we evaluate a multivariable family of infinite products which generalize Guillera's infinite product for $e$, and Ser's formula (rediscovered by Sondow) for $e^γ$. We describe formulas for the products in terms of special values of the Hurwitz zeta function $ζ(s,u)$ and its $s$ derivative. Additionally, we derive integral and double integral representations for the logarithms of these infinite products.

On Product Formulas of Guillera and Sondow

Abstract

In this note, we evaluate a multivariable family of infinite products which generalize Guillera's infinite product for , and Ser's formula (rediscovered by Sondow) for . We describe formulas for the products in terms of special values of the Hurwitz zeta function and its derivative. Additionally, we derive integral and double integral representations for the logarithms of these infinite products.

Paper Structure

This paper contains 5 sections, 6 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Proofs
  4. Proof of Main Theorem
  5. Proof of Integral Formulas

Key Result

Theorem 1

For $u>0$ and each integer $d\geq 0$, we have Consequently,

Theorems & Definitions (20)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Example 1
  • Example 2
  • Corollary 3
  • proof
  • Example 3
  • ...and 10 more