Asymptotic expansion of orthogonal polynomials via difference equations (Q1717622)

This paper deals with the asymptotic solution of the linear difference equation \[ P_{n+1}(x)+P_{n-1}(x)=(A_nx-B_n) P_n(x),\quad n\geq 1, \] where \[ A_n\sim n^{-\theta}\sum_{k=0}^\infty \frac{\alpha_k}{n^k},\quad B_n\sim \sum_{k=0}^\infty \frac{\beta_k}{n^k}\qquad(n\to\infty) \] with \(\theta>0\) and \(\alpha_0>0\). Many well-known orthogonal polynomials (such as the Hermite, Laguerre and Meixner polynomials) satisfy the above difference equation. By means of a logarithm-type procedure, the authors exploit the symmetry in the difference equation to determine an asymptotic expansion up to any order of two linearly independent solutions valid away from turning points. This procedure is combined with the method of asymptotic matching to determine the expansions in the outer and oscillatory regions. To illustrate this approach the authors use the Hermite and continuous dual Hahn polynomials as examples. In the case of the Hermite polynomials a numerical verification of the accuracy of the expansion developed is given.