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

TL;DR

This paper explores difference and differential relations for orthogonal polynomials in the Askey scheme using quantum mechanical formulations and shape invariance properties.

Contribution

It introduces new difference and differential relations for these polynomials based on the shape invariance and shift relations in quantum mechanics.

Findings

Derived difference relations for Askey scheme polynomials using quantum formulations.

Established differential relations for classical orthogonal polynomials like Jacobi.

Connected multiplication by a polynomial to surjective maps between Hilbert spaces.

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 $\mathsf{H}_{\lambda}$ ($\lambda$ : a set of parameters) and that with shifted parameters $\mathsf{H}_{\lambda+\delta}$ ($\delta$ : shift of $\lambda$), 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 $\check{\Phi}(x)$, 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 $\sqrt{\check{\Phi}(x)}$ gives a surjective map from $\mathsf{H}_{\lambda+2\delta}$ to $\mathsf{H}_{\lambda}$. 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 $\sqrt{\check{\Phi}(x)}$ in this case gives a surjective map from $\mathsf{H}_{\lambda+\delta}$ to $\mathsf{H}_{\lambda}$.