Plancherel_Rotach Asymptotics for q-Orthogonal Polynomials
Mourad E. H. Ismail, Xin Li

TL;DR
This paper derives Plancherel-Rotach asymptotics near the largest zero for certain q-orthogonal polynomials, including q-Hermite, q-Laguerre, and Stieltjes-Wigert, expanding understanding of their asymptotic behavior.
Contribution
It provides new asymptotic formulas at the soft edge for classes of q-orthogonal polynomials and introduces a one-parameter family of solutions to the Ramanujan q-difference equation.
Findings
Asymptotics established for q-Hermite, q-Laguerre, and Stieltjes-Wigert polynomials.
New solutions to the Ramanujan q-difference equation introduced.
Enhanced understanding of the behavior of q-orthogonal polynomials near their largest zeros.
Abstract
We establish the Plancherel-Rotach-type asymptotics around the largest zero (the soft edge asymptotics) for some classes of polynomials satisfying three-term recurrence relations with exponentially increasing coefficients. As special cases, our results include this type of asymptotics for q^{-1}-Hermite polynomials of Askey, Ismail and Masson, q-Laguerre polynomials, and the Stieltjes-Wigert polynomials. We also introduce a one parameter family of solutions to the q-difference equation of the Ramanujan function.
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Taxonomy
TopicsMathematical functions and polynomials · Advanced Mathematical Identities · Advanced Combinatorial Mathematics
