Recurrence equations and their classical orthogonal polynomial solutions
Classical orthogonal polynomials of a continuous or a discrete variable can be classified using second order differential or (\(q\)-)difference equations of the following form \[ \begin{aligned} \sigma(x)y''(x)+\tau(x)y'(x)+\lambda_ny(x)&=0,\\ \sigma(x)\Delta\nabla y(x)+\tau(x)\Delta y(x)+\lambda_ny(x)&=0,\\ \sigma(x)D_qD_{1/q} y(x)+\tau(x) D_q y(x)+\lambda_{q,n}y(x)&=0.\end{aligned} \] Here \(\sigma(x), \tau(x)\) are polynomials of at most degree two and one respectively. The difference operators are defined by \[ \Delta y(x)=y(x+1)-y(x),\;\nabla y(x)=y(x)-y(x-1),\;D_qf(x)={f(qx)-f(x)\over (q-1)x},\;q\not= 1. \] The authors now give a general method to express the coefficients \(A_n, B_n\) and \(C_n\) of the homogeneous recurrence equation \[ p_{n+1}(x)=(A_nx+B_n)p_n(x)-C_np_{n-1}(x), \] satisfied by the classical orthogonal polynomials, in terms of the given polynomial coefficients of the differential/difference equation. The classification theorems give explicit results and in three tables all the classical (and some `degenerate') cases are given, along with algorithms using Maple. For the implementation of the algorithms (retode) as well as a worksheet with the examples from the paper (\texttt{retode.mws}), the reader is referred to the web-page of the first author (\url{http://www.mathematik.uni-kassel.de/~koepf/Publikationen}). Also the authors discuss the differences with the computer program \texttt{rec2ortho} as implemented by Koornwinder and Swarttouw (\url{http://turing.wins.uva.nl/~thk/recentpapers/rec2ortho.html}) in 1996-1998. A very nicely written paper.
scientific article; zbMATH DE number 1861075
scientific article