Generic formulas for the values at the singular points of some special monic classical [formula omitted]-orthogonal polynomials

Abstract

It is well-known that the classical orthogonal polynomials of Jacobi, Bessel, Laguerre and Hermite are solutions of a Sturm–Liouville problem of the type σ ( x ) y n ″ + τ ( x ) y n ′ - λ n y n = 0 , where σ and τ are polynomials such that deg σ ⩽ 2 and deg τ = 1 , and λ n is a constant independent of x. Recently, based on the hypergeometric character of the solutions of this differential equation, W. Koepf and M. Masjed-Jamei [A generic formula for the values at the boundary points of monic classical orthogonal polynomials, J. Comput. Appl. Math. 191 (2006) 98–105] found a generic formula, only in terms of the coefficients of σ and τ , for the values of the classical orthogonal polynomials at the singular points of the above differential hypergeometric equation. In this paper, we generalize the mentioned result giving the analogous formulas for both the classical q -orthogonal polynomials (of the q -Hahn tableau) and the classical D ω -orthogonal polynomials. Both are special cases of the classical H q , ω -orthogonal polynomials, which are solutions of the hypergeometric-type difference equation σ ( x ) H q , ω H 1 / q ,- ω y n + τ ( x ) H q , ω y n - λ n y n = 0 , where H q , ω is the difference operator introduced by Hahn, and σ , τ and λ n being as above. Our approach is algebraic and it does not require hypergeometric functions.