Differential, difference, and asymptotic relations for Pollaczek–Jacobi type orthogonal polynomials and their Hankel determinants

Abstract

AbstractIn this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek–Jacobi type weight urn:x-wiley:00222526:media:sapm12392:sapm12392-math-0001By using the ladder operator approach, we establish the second‐order difference equations satisfied by the recurrence coefficient and the sub‐leading coefficient of the monic orthogonal polynomials, respectively. We show that the logarithmic derivative of can be expressed in terms of a particular Painlevé V transcendent. The large asymptotic expansions of and are obtained by using Dyson's Coulomb fluid method together with the related difference equations. Furthermore, we study the associated Hankel determinant and show that a quantity , allied to the logarithmic derivative of , can be expressed in terms of the ‐function of a particular Painlevé V. The second‐order differential and difference equations for are also obtained. In the end, we derive the large asymptotics of and from their relations with and .