On properties of a deformed Freud weight

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.