A Generalization of the Barnes $G$-Function

Abstract

Generalizing the Barnes $G$-function we define an entire function of order $m$ with the zeros $k$ with multiplicity $k^m(k,m\in \mathbb{N})$. We prove functional equations for it and study its asymptotic behaviour and Taylor series. This generalization is useful in order to describe the topological zeros of the Selberg zeta function with respect to the spectrum of the Laplace operator for differential $p$-forms on $n$-dimensional compact hyperbolic space forms.