Some Difference Relations for Orthogonal Polynomials of a Continuous Variable in the Askey Scheme

TL;DR

The paper develops a unified, algebraic approach to derive difference and differential relations for orthogonal polynomials in the Askey scheme by exploiting shape invariance in idQM and oQM. Central to the method is the polynomial modifier $\check{\Phi}(x;\bm{\lambda})$ that links shifted and unshifted Hilbert spaces via Christoffel transformations, enabling explicit forward/backward shift relations and the resulting difference equations for polynomials such as the Askey–Wilson family and Jacobi polynomials. The authors provide detailed AW (and J) explicit expressions for the coefficients and zeros involved in these relations, and show how a surjective map given by $\sqrt{\check{\Phi}(x;\bm{\lambda})}$ connects $\mathsf{H}_{\bm{\lambda}+2\bm{\delta}}$ to $\mathsf{H}_{\bm{\lambda}}$ (and similarly for oQM with $\bm{\delta}$). The framework extends to one-parameter shifts and offers a path toward multi-indexed or exceptional polynomial families, highlighting broad applicability and potential future developments in the theory of orthogonal polynomials and their quantum-mechanical interpretations.

Abstract

Orthogonal polynomials of a continuous variable in the Askey scheme satisfying second order difference equations, such as the Askey-Wilson polynomial, can be studied by the quantum mechanical formulation, idQM (discrete quantum mechanics with pure imaginary shifts). These idQM systems have the shape invariance property, which relates the Hilbert space $\mathsf{H}_λ$ ($λ$ : a set of parameters) and that with shifted parameters $\mathsf{H}_{λ+δ}$ ($δ$ : shift of $λ$), and gives the forward and backward shift relations for the orthogonal polynomials. Based on the forward shift relation and the Christoffel's theorem with some polynomial $\checkΦ(x)$, which is expressed in terms of the quantities appeared in the forward and backward shift relations, we obtain some difference relations for the orthogonal polynomials. The multiplication of $\sqrt{\checkΦ(x)}$ gives a surjective map from $\mathsf{H}_{λ+2δ}$ to $\mathsf{H}_λ$. Similarly, for the orthogonal polynomials in the Askey scheme satisfying second order differential equations, such as the Jacobi polynomial, we obtain some differential relations, and the multiplication of $\sqrt{\checkΦ(x)}$ in this case gives a surjective map from $\mathsf{H}_{λ+δ}$ to $\mathsf{H}_λ$.