Plancherel-Rotach Asymptotics for Ismail-Masson Orthogonal Polynomials with Complex Scaling

Plancherel_Rotach Asymptotics for q-Orthogonal Polynomials

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.

Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems

We study the Plancherel--Rotach asymptotics of four families of orthogonal polynomials, the Chen--Ismail polynomials, the Berg-Letessier-Valent polynomials, the Conrad--Flajolet polynomials I and II. All these polynomials arise in indeterminate moment problems and three of them are birth and death process polynomials with cubic or quartic rates. We employ a difference equation asymptotic technique due to Z. Wang and R. Wong. Our analysis leads to a conjecture about large degree behavior of polynomials orthogonal with respect to solutions of indeterminate moment problems.

Plancherel-Rotach Asymptotics for $q$-Series

In this work we study the Plancherel-Rotach type asymptotics for selected $q$-series and $q$-orthogonal polynomials with complex scalings. The $q$-series we cover are Euler's $q$-exponential, Ramanujan function, Jackson's $q$-Bessel function of second kind, Ismail-Masson orthogonal polynomials, Stieltjes-Wigert polynomials and $q$-Laguerre polynomials. For a fixed $q$ with $0<q<1$, in each case the main term of the asymptotic formulas may contain Ramanujan function or theta function depending on the value of scaling parameter. Furthemore, when the scaling parameter is in certain strip of the c…

Scaled Asymptotics For Some $q$-Series As $q$ Approaching Unit

In this work we investigate Plancherel-Rotach type asymptotics for some $q$-series as $q\to1$. These $q$-series generalize Ramanujan function $A_{q}(z)$; Jackson's $q$-Bessel function $J_ν^{(2)}$(z;q), Ismail-Masson orthogonal polynomials($q^{-1}$-Hermite polynomials) $h_{n}(x|q)$, Stieltjes-Wigert orthogonal polynomials $S_{n}(x;q)$, $q$-Laguerre orthogonal polynomials $L_{n}^{(α)}(x;q)$ and confluent basic hypergeometric series.

Some orthogonal very-well-poised $_8φ_7$-functions that generalize Askey-Wilson polynomials

In a recent paper Ismail, Masson, and Suslov have established a continuous orthogonality relation and some other properties of a $_2φ_1$-Bessel function on a $q$-quadratic grid. Dick Askey suggested that the ``Bessel-type orthogonality'' at the $_2φ_1$-level has really a general character and can be extended up to the $_8φ_7$-level. Very-well-poised $_8φ_7$-functions are known as a nonterminating version of the classical Askey--Wilson polynomials. Askey's congecture has been proved by the author. In the present paper we discuss in details some properties of the orthogonal $_8φ_7$-functions. An…

类型 $R_{II}$ 的广义正交多项式的组合学

Combinatorics of generalized orthogonal polynomials of type $R_{II}$

本文将Kim和Stanton关于类型R_I正交多项式矩的组合模型推广到类型R_{II}，并通过放宽条件定义广义类型R_{II}正交多项式，证明了一个统一多种正交多项式矩组合模型的主定理。