Another type of forward and backward shift relations for orthogonal polynomials in the Askey scheme
The (basic) hypergeometric orthogonal polynomials in the Askey scheme satisfy second-order differential or difference equations and the forward and backward shift relations are their basic properties. The orthogonal polynomials in the Askey scheme provide us with exactly solvable quantum mechanical models.\N\NConversely, one can use the quantum mechanical formulation as a tool to investigate orthogonal polynomials, see [\textit{S. Odake} and \textit{R. Sasaki}, J. Phys. A, Math. Theor. 44, No. 35, Article ID 353001, 47 p. (2011; Zbl 1227.81167)]. For example, the forward and backward shift relations result from the shape invariance, and the multi-indexed orthogonal polynomials are found using the quantum mechanical formulation.\N\NThe forward and backward shift relations are related to the factorization of the Hamiltonian. This paper investigates whether such new factorization and forward and backward shift relations exist for other orthogonal polynomials. In addition to the finite discrete quantum mechanics with real shift systems (\(q\)-Racah, etc.), they examine the ordinary quantum mechanics systems (Jacobi, etc.), the discrete quantum mechanics with imaginary shift systems (Askey-Wilson, etc.), the semi-infinite discrete quantum mechanics with real shift systems (\(q\)-Meixner, etc.) and the discrete quantum mechanics with real shift systems with the Jackson integral type measure (big \(q\)-Jacobi, etc.). Essentially, these shift relations shift only the parameters.
scientific article; zbMATH DE number 7915172
scientific article