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}$}$inwhichthebirthanddeathrates$\lambda_n=q^{2\nu-2n}$and$\mu_n=q^{-2n}$where$0<q<1$and$\nu>-1$ . The mathematical methods employed here are based on $q$-Bessel Fourier analysis.