# Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations

Clarkson, Peter (2013) Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations. Journal of Physics A: Mathematical and Theoretical, 46 (18). ISSN 1751-8113. (doi:10.1088/1751-8113/46/18/185205) (KAR id:33120)

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http://dx.doi.org/10.1088/1751-8113/46/18/185205

## Abstract

We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations can be expressed in terms of Wronskians of modified Bessel functions and confluent hypergeometric functions, respectively for the generalized Charlier and generalized Meixner polynomials. These Wronskians arise in the description of special function solutions of the third and fifth Painlevé equations.

Item Type: Article 10.1088/1751-8113/46/18/185205 Q Science > QA Mathematics (inc Computing science) > QA351 Special functionsQ Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Applied Mathematics Peter Clarkson 30 Jan 2013 18:55 UTC 01 Aug 2019 10:35 UTC https://kar.kent.ac.uk/id/eprint/33120 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-8777-5284