# Linear Difference Equations with a Transition Point at the Origin

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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 $\theta eq0$ and $\alpha 0 eq0$ 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 $J u$ and $Y u$ 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 $x^\alpha\exp-q mx^m$, $x0$, where $m$ is a positive integer, $\alpha-1$ and $q m0$.

Autor: Lihua Cao; Yutian Li

Fuente: https://archive.org/