# A Generalized Freud Weight

Clarkson, Peter, Jordaan, Kerstin, Kelil, Abey (2015) A Generalized Freud Weight. Studies in Applied Mathematics, 136 (3). pp. 288-320. ISSN 0022-2526. E-ISSN 1467-9590. (doi:10.1111/sapm.12105)

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http://dx.doi.org/10.1111/sapm.12105

## Abstract

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a generalized Freud weight w(x;t)=|x|2?+1exp(?x4+tx2),x?R, with parameters ?>?1 and t?R, and classical solutions of the fourth Painlev\'e equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the fourth Painlev\'e equation. Further we derive a second-order linear ordinary differential equation and a differential-difference equation satisfied by the generalized Freud polynomials.

Item Type: Article 10.1111/sapm.12105 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 A Clarkson 20 Oct 2015 10:03 UTC 29 May 2019 16:10 UTC https://kar.kent.ac.uk/id/eprint/51095 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-8777-5284