Recurrence equations and their classical orthogonal polynomial solutions

• Authors: Wolfram Koepf;Dieter Schmersau
• Venue: Applied Mathematics and Computation - Orthogonal systems and applications
• Year: 2002

The classical orthogonal polynomials are given as the polynomial solutions pn(x) of the differential equation σ(x)y''(x) + τ(x)y'(x) + λy(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 equation pn+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 equation σ(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 equation σ(x)DqD1/qy(x) + τ(x)Dqy(x) + λq,ny(x) = 0, where Dqf(x) = f(qx)-f(x)/(q-1)x , q ≠ 1, denotes the q-difference operator.