Recurrence Coefficients of a New Generalization of the Meixner Polynomials

Abstract

We investigate new generalizations of the Meixner polynomials on the lattice N, on the shifted lattice N+1−β and on the bi-lattice N∪(N+1−β). We show that the coefficients of the three-term recurrence relation for the orthogonal polynomials are related to the solutions of the fifth Painlevé equation P_V. Initial conditions for different lattices can be transformed to the classical solutions of P_V with special values of the parameters. We also study one property of the Bäcklund transformation of P_V.

Keywords

• Painlevé equations
• Bäcklund transformations
• classical solutions
• orthogonal polynomials
• recurrence coefficients