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

Abstract

The classical Lerch's formula states the following normalized product: $$\prod _{n=0}^{\infty }(x+n)=\frac{\sqrt{2\pi }}{\Gamma (x)},\text{Re}(x)>0,$$ 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 $$\frac{2\cdot 2}{1\cdot 3}\frac{4\cdot 4}{3\cdot 5}\frac{6\cdot 6}{5\cdot 7}\cdots =\frac{\pi }{2}$$ (Wallis' formula in 1656), respectively.

10 pages

Cited by in corpus (1)

• Mizuno-type result and Wallis' formula