Some Difference Relations for Orthogonal Polynomials of a Continuous Variable in the Askey Scheme
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 $\mathsf{H}_{\lambda}$.