Orthogonal polynomials on a bi-lattice

Abstract

We investigate generalizations of the Charlier and the Meixner polynomials on the lattice N and on the shifted lattice N+1-β. We combine both lattices to obtain the bi-lattice N \cup (N+1-β) and show that the orthogonal polynomials on this bi-lattice have recurrence coefficients which satisfy a non-linear system of recurrence equations, which we can identify as a limiting case of an (asymmetric) discrete Painlevé equation.

25 pages, 2 figures

References in corpus (3)

• Painleve IV and degenerate Gaussian Unitary Ensembles
• Orthogonal polynomials with discontinuous weights
• Painlevé V and the distribution function of discontinuous linear statistics in the Laguerre Unitary Ensembles

Cited by in corpus (12)

• The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation
• Discrete semiclassical orthogonal polynomials of class one
• Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation
• Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations
• Recurrence Coefficients of a New Generalization of the Meixner Polynomials
• Orthogonal and multiple orthogonal polynomials, random matrices, and Painlevé equations
• Classical Solutions of the Degenerate Fifth Painlevé Equation
• Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlevé VI
• The generalized Krawtchouk polynomials and the fifth Painlevé equation
• Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials
• Recurrence relations for the generalized Laguerre and Charlier orthogonal polynomials and discrete Painlevé equations on the $D_6^{(1)}$ Sakai surface
• Discrete multiple orthogonal polynomials on shifted lattices