Recurrence equations and their classical orthogonal polynomial solutions

摘要

The classical orthogonal polynomials are given as the polynomial solutions pn(x) of the differential equationσ(x)y″(x)+τ(x)y′(x)+λny(x)=0,where σ(x) is a polynomial of at most second degree and τ(x) is a polynomial of first degree.In this paper a general method to express the coefficients An,Bn and Cn of the recurrence equationpn+1(x)=(Anx+Bn)pn(x)−Cnpn−1(x)in terms of the given polynomials σ(x) and τ(x) is used to present an algorithm to determine the classical orthogonal polynomial solutions of any given holonomic three-term recurrence equation, i.e., a homogeneous linear three-term recurrence equation with polynomial coefficients.In a similar way, classical discrete orthogonal polynomial solutions of holonomic three-term recurrence equations can be determined by considering their corresponding difference equationσ(x)Δ∇y(x)+τ(x)Δy(x)+λny(x)=0,where Δy(x)=y(x+1)−y(x) and ∇y(x)=y(x)−y(x−1) denote the forward and backward difference operators, respectively, and a similar approach applies to classical q-orthogonal polynomials, being solutions of the q-difference equationσ(x)DqD1/qy(x)+τ(x)Dqy(x)+λq,ny(x)=0,whereDqf(x)=f(qx)−f(x)(q−1)x,q≠1,denotes the q-difference operator.