Generalizations of Lerch's formula by Barnes' multiple zeta functions

TL;DR

This paper generalizes Lerch's formula using Barnes' multiple zeta functions, leading to new product identities including the regularized product of natural numbers and Wallis' formula.

Contribution

It introduces two new generalizations of Lerch's formula via Barnes' multiple zeta functions, connecting to classical product formulas.

Findings

Derived a generalized product formula for natural numbers using Barnes' zeta functions.

Established a new form of Wallis' product through these generalizations.

Connected classical formulas with Barnes' multiple zeta functions.

Abstract

The classical Lerch's formula states the following normalized product: $$\prod_{n=0}^\infty(x+n)=\frac{\sqrt{2\pi}}{\Gamma(x)},\quad \textrm{Re}(x)>0, $$ where $\Gamma(x)$ is the Euler gamma function. In this note, by using Barnes' multiple zeta function and its alternating form, we obtain two kinds of generalizations of Lerch's formula, which imply the product $$ \prod_{n=1}^\infty n=\sqrt{2\pi} $$ (in the sense of zeta regularization) and the product $$\frac{2\cdot2}{1\cdot 3}\frac{4\cdot4}{3\cdot 5}\frac{6\cdot6}{5\cdot 7}\cdots=\frac{\pi}{2}$$ (Wallis' formula in 1656), respectively.