Comparative asymptotics for discrete semiclassical orthogonal polynomials

Abstract

We study the ratio P n ( x ; z ) φ n ( x ) asymptotically as n → ∞ , where the polynomials P n ( x ; z ) are orthogonal with respect to a discrete linear functional and φ n ( x ) denote the falling factorial polynomials. We give recurrences that allow the computation of high order asymptotic expansions of P n ( x ; z ) and give examples for most discrete semiclassical polynomials of class s ≤ 2 . We show several plots illustrating the accuracy of our results.