Bilateral Birth and death process in quantum calculus

TL;DR

This paper provides a complete solution to bilateral birth and death processes in quantum calculus using q-Bessel Fourier analysis, focusing on specific q-dependent birth and death rates.

Contribution

It introduces a novel analytical approach to solving bilateral birth and death processes in quantum calculus with explicit solutions.

Findings

Explicit solutions for the bilateral birth and death process equations.

Application of q-Bessel Fourier analysis to stochastic processes.

Characterization of the process dynamics in quantum calculus setting.

Abstract

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