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 PV. Initial conditions for different lattices can be transformed to the classical solutions of PV with special values of the parameters. We also study one property of the Bäcklund transformation of PV.