Orthogonal polynomials and second-order pseudo-spectral linear differential equations

Abstract

In this paper, we deal with monic orthogonal polynomial sequences which satisfy the second-order pseudo-spectral linear differential equation: where φ i , i=1, 2 are polynomials with φ2 monic, and the degrees of the polynomials χ(·, n) are uniformly bounded. These polynomial sequences are semiclassical of class either s=0 or 1. They are, up to a linear change of variable, the classical polynomials (Hermite, Laguerre, Bessel, and Jacobi) and symmetric semiclassical polynomials of class one. For them, we deduce the three-term recurrence relations, the structure relations, and the second-order linear differential equations that these polynomial sequences satisfy.