Linear Difference Equations with a Transition Point at the Origin

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^{-θ}\sum_{s=0}^\infty\frac{α_s}{n^s},\qquad B_n\sim\sum_{s=0}^\infty\frac{β_s}{n^s}, {equation*} with $θ\neq0$ and $α_0\neq0$ being real numbers, and $β_0=\pm2$. Our result hold uniformly for the scaled variable $t$ in an infinite interval containing the transition point $t_1=0$, where $t=(n+τ_0)^{-θ} x$ and $τ_0$ is a small shift. In particular, it is shown how the Bessel functions $J_ν$ and $Y_ν$ 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 $x^α\exp(-q_mx^m)$, $x>0$, where $m$ is a positive integer, $α>-1$ and $q_m>0$.

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