Birth and death processes associated with little $q-$Laguerre orthogonal polynomials.

Abstract

We study birth and death processes with rates $\lambda_{n}=q^{n}(1-aq^{n+1}),~ \mu_{n}=aq^{n}(1-q^{n}), ~ n\ge 0$ ~and $0<a,q<1 $. Using the associated generating functions, we show that corresponding orthogonal polynomials generalize little $q$-Laguerre polynomials. We also give the minimal solution of the three-term recurrence relation, and we obtain some formulas for the convergent of continued fractions associated with the little $q$-Laguerre orthogonal polynomials.