Linear Difference Equations with a Transition Point at the Origin

Abstract

A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation {equation*} P_{n+1}(x)-(A_{n}x+B_{n})P_{n}(x)+P_{n-1}(x)=0, {equation*} where $A_n$ and $B_n$ have asymptotic expansions of the form {equation*} A_n\sim n^{-\theta}\sum_{s=0}^\infty\frac{\alpha_s}{n^s},\qquad B_n\sim\sum_{s=0}^\infty\frac{\beta_s}{n^s}, {equation*} with $\theta\neq0$ and $\alpha_0\neq0$ being real numbers, and $\beta_0=\pm2$. Our result hold uniformly for the scaled variable $t$ in an infinite interval containing the transition point $t_1=0$, where $t=(n+\tau_0)^{-\theta} x$ and $\tau_0$ is a small shift. In particular, it is shown how the Bessel functions $J_\nu$ and $Y_\nu$ get involved in the uniform asymptotic expansions of the solutions to the above three-term recurrence relation. As an illustration of the main result, we derive a uniform asymptotic expansion for the orthogonal polynomials associated with the Laguerre-type weight $x^\alpha\exp(-q_mx^m)$, $x>0$, where $m$ is a positive integer, $\alpha>-1$ and $q_m>0$.