Recurrence Relations of the Multi-Indexed Orthogonal Polynomials V : Racah and $q$-Racah types

Abstract

In previous papers, we discussed the recurrence relations of the multi-indexed orthogonal polynomials of the Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we explore those of the Racah and $q$-Racah types. For the $M$-indexed ($q$-)Racah polynomials, we derive $3+2M$ term recurrence relations with variable dependent coefficients and $1+2L$ term ($L\ge M+1$) recurrence relations with constant coefficients. Based on the latter, the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi-indexed ($q$-)Racah polynomials are obtained. In appendix we present a proof and some data of the recurrence relations with constant coefficients for the multi-indexed Wilson and Askey-Wilson polynomials.

45 pages. Comments are added. To appear in JMP

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Cited by in corpus (3)

• Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types
• New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types