A generalized sextic Freud weight. Issue 5 (3rd August 2021)

• Record Type: Journal Article
• Authors: Clarkson, Peter A.; Jordaan, Kerstin
• Is Part Of: Integral transforms and special functions. Volume 32:Issue 5/8(2021)
• Page Start: 458
• Page End: 482
• Publication Date: 2021-08-03
• Subjects: Generalized sextic freud weight -- generalized hypergeometric functions -- semi-classical orthogonal polynomials -- recurrence coefficients -- discrete Painlevé equation -- Painlevé equation; 33C20 -- 33C47 -- 34M55 -- 65Q99; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2020.1798949
• Languages: English
• ISSNs: 1065-2469

Abstract

ABSTRACT: We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalized sextic Freud weight 1 ω ( x ; t, λ ) = | x | 2 λ + 1 exp − x 6 + t x 2, x ∈ R, with parameters λ > − 1 and t ∈ R . We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of generalized hypergeometric functions 1 F 2 ( a 1 ; b 1, b 2 ; z ) . We derive a nonlinear discrete as well as a system of differential equations satisfied by the recurrence coefficients and use these to investigate their asymptotic behaviour. We conclude by highlighting a fascinating connection between generalized quartic, sextic, octic and decic Freud weights when expressing their first moments in terms of generalized hypergeometric functions.