Bilateral Birth and death process in quantum calculus

Abstract

In this paper I shall give the complete solution of the equations governing the bilateral birth and death process on path set ${\mathbb{R}}_q=\{q^n,n\in \mathbb{Z}\}$ in which the birth and death rates ${\lambda }_n=q^{2\nu -2n}$ and ${\mu }_n=q^{-2n}$ where $0<q<1$ and $\nu >-1$ . The mathematical methods employed here are based on $q$-Bessel Fourier analysis.