2015/10/28 by Peter A. Clarkson, Kerstin Jordaan, Abey Kelil · 2 citations
Mathematics · #math.CA #msc:34M55 #msc:33E17 #msc:33C47
paper · pdf · doi:10.1111/sapm.12105
published as Studies in Applied Mathematics, 136 (2016) 288-320 · 22 pages, Studies in Applied Mathematics, accepted for publication
arxiv created 2015/10/28 · arxiv updated 2017/11/07
We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a generalized Freud weight w(x;t)=|x|2λ+1exp(-x4+tx2), x∈ℝ, with parameters λ>-1 and t∈ℝ, and classical solutions of the fourth Painlevé 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 Painlevé equation. Further we derive a second-order linear ordinary differential equation and a differential-difference equation satisfied by the generalized Freud polynomials.