Affiliation:
1. Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, P. R. China
Abstract
We study the recurrence coefficients of the monic polynomials [Formula: see text] orthogonal with respect to the deformed (also called semi-classical) Freud weight [Formula: see text] with parameters [Formula: see text]. We show that the recurrence coefficients [Formula: see text] satisfy the first discrete Painlevé equation (denoted by d[Formula: see text]), a differential–difference equation and a second-order nonlinear ordinary differential equation (ODE) in [Formula: see text]. Here [Formula: see text] is the order of the Hankel matrix generated by [Formula: see text]. We describe the asymptotic behavior of the recurrence coefficients in three situations: (i) [Formula: see text], [Formula: see text] finite, (ii) [Formula: see text], [Formula: see text] finite, (iii) [Formula: see text], such that the radio [Formula: see text] is bounded away from [Formula: see text] and closed to [Formula: see text]. We also investigate the existence and uniqueness for the positive solutions of the d[Formula: see text]. Furthermore, we derive, using the ladder operator approach, a second-order linear ODE satisfied by the polynomials [Formula: see text]. It is found as [Formula: see text], the linear ODE turns to be a biconfluent Heun equation. This paper concludes with the study of the Hankel determinant, [Formula: see text], associated with [Formula: see text] when [Formula: see text] tends to infinity.
Funder
Macau Science and Technology Development Fund
Universidade de Macau (MO)
Universidade de Macau
Publisher
World Scientific Pub Co Pte Lt
Subject
Discrete Mathematics and Combinatorics,Statistics, Probability and Uncertainty,Statistics and Probability,Algebra and Number Theory
Cited by
12 articles.
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