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Asymptotics for some $q$-hypergeometric polynomials

Juan F. Mañas-Mañas, Juan J. Moreno-Balcázar

TL;DR

This work extends local Mehler–Heine asymptotics to a broad class of $q$-hypergeometric polynomials $_s\phi_s$ by a delicate $q$-dependent scaling of the argument, linking the limits to the $q$-Bessel function $J^{(2)}_{\alpha-1}$. The main result provides an explicit uniform limit for $s\ge 2$ (and a special case $s=1$) and discusses extensions to $r-1\le s$, with a detailed analysis of the required bounds via the $q$-Gamma function and $q$-Pochhammer symbols. The paper also demonstrates the consequences for the zeros through Hurwitz’s theorem, illustrates the theory with two classical examples ($q$-Laguerre and Little-$q$-Jacobi polynomials), and presents numerical experiments that reveal rich zero-structure dependent on $\alpha$ and $q$, while underscoring open questions about the zeros of the limiting $q$-Bessel function.

Abstract

We tackle the study of a type of local asymptotics, known as Mehler--Heine asymptotics, for some $q$--hypergeometric polynomials. Some consequences about the asymptotic behavior of the zeros of these polynomials are discussed. We illustrate the results with numerical examples.

Asymptotics for some $q$-hypergeometric polynomials

TL;DR

This work extends local Mehler–Heine asymptotics to a broad class of -hypergeometric polynomials by a delicate -dependent scaling of the argument, linking the limits to the -Bessel function . The main result provides an explicit uniform limit for (and a special case ) and discusses extensions to , with a detailed analysis of the required bounds via the -Gamma function and -Pochhammer symbols. The paper also demonstrates the consequences for the zeros through Hurwitz’s theorem, illustrates the theory with two classical examples (-Laguerre and Little--Jacobi polynomials), and presents numerical experiments that reveal rich zero-structure dependent on and , while underscoring open questions about the zeros of the limiting -Bessel function.

Abstract

We tackle the study of a type of local asymptotics, known as Mehler--Heine asymptotics, for some --hypergeometric polynomials. Some consequences about the asymptotic behavior of the zeros of these polynomials are discussed. We illustrate the results with numerical examples.
Paper Structure (5 sections, 12 theorems, 80 equations)

This paper contains 5 sections, 12 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Technical results
  3. Main result
  4. Two classical examples
  5. Discussion about the zeros

Key Result

Lemma 1

Let $k$ be a nonnegative integer and $z$ a complex number, such that $\Gamma_{q}(z)$ is well defined, then

Theorems & Definitions (30)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 20 more