Bilateral Birth and death process in quantum calculus

In this paper I shall give the complete solution of the equations governing the bilateral birth and death process on path set $\mathbb{R}_q=\{q^n,\quad n\in\mathbb{Z}\}$ 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.