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

In this paper, we deal with monic orthogonal polynomial sequences which satisfy the second-order pseudo-spectral linear differential equation: phi(2)((x)y ''(x) + phi(1)(x)y'x) = chi(x, n)y(x), n is an element of N, where phi(i), i = 1, 2 are polynomials with phi(2) monic, and the degrees of the polynomials chi(., 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.