Properties of generalized Freud polynomials

Abstract

We consider the semiclassical generalized Freud weight functionWe analyse the asymptotic behaviour of the sequences of monic polynomials that are orthogonal with respect to w λ (x; t), as well as the asymptotic behaviour of the recurrence coefficient, when the degree, or alternatively, the parameter t, tend to infinity. We also investigate existence and uniqueness of positive solutions of the nonlinear discrete equation satisfied by the recurrence coefficients and prove properties of the zeros of the generalized Freud polynomials.

The Symmetric Semi-classical Orthogonal Polynomials of Class Two and Some of Their Extensions

“…with Q an even, non-negative and continuous real valued function defined on the real line (satisfying certain conditions involving its first and second derivatives), are nowadays commonly known as Freud-type weights. The cases Q(x) = |x| m , m ∈ N, have been extensively studied, main references and results can be found in the introduction section of [11]. A common topic of research concerns the derivation and study of the systems of nonlinear difference equations satisfied by the recurrence relation coefficients of the corresponding orthogonal polynomials.…”

“…Many other examples of discrete Painlevé equations for the recurrence relation coefficients of orthogonal polynomials have been studied (see [6,14,16,24]). Applications of Laguerre-Freud equations to the study of asymptotics for the orthogonal polynomials, properties of zeroes, estimates for derivatives, inequalities, etc, can be found in a vast list of references (see, amongst many others, [2,11,32] and its lists of references).…”

On a class of q -orthogonal polynomials and the q -Riemann–Hilbert problem

“…It is now known that many families of orthogonal polynomials have recurrence coefficients determined by additive-type discrete Painlevé equations [10,19,20]. More recently, it has been shown that the recurrence coefficients of q-orthogonal polynomials satisfy multiplicative-type discrete Painlevé equations [21], where now the non-autonomous term in the equation is iterated on multiplicative lattices.…”

On properties of a deformed Freud weight

“…(23), P. 5], written by Freud, see also [1, §2]. Joshi and Lustri [25] studied the large n behavior of dP I , see also [13].…”