相关论文: Plancherel-Rotach Asymptotics for $q$-Series

We study the Lambert series $\mathscr{L}_q(s,x) = \sum_{k=1}^\infty k^s q^{k x}/(1-q^k)$, for all $s \in \mathbb{C}$. We obtain the complete asymptotic expansion of $\mathscr{L}_q(s,x)$ near $q=1$. Our analysis of the Lambert series yields…

We describe the strong asymptotic behavior of type I and type II multiple orthogonal polynomials which were used to give an integral expression for a transition probability function corresponding to a queueing models that has a bulk service…

We analyze the large degree asymptotic behavior of matrix valued orthogonal polynomials (MVOPs), with a weight that consists of a Jacobi scalar factor and a matrix part. Using the Riemann-Hilbert formulation for MVOPs and the Deift-Zhou…

Diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra U_q(su(1,1)) is studied. Spectrum and eigenfunctions of this operator are found in an explicit form. These…

We study two families of orthogonal polynomials. The first is a finite family related to the Askey-Wilson polynomials but the orthogonality is on the real line. A limiting case of this family is an infinite system of orthogonal polynomials…

Recently, $\lambda$-Bernoulli and $\lambda$-Euler numbers are studied in [5, 10]. The purpose of this paper is to present a systematic study of some families of the $q$-extensions of the $\lambda$-Bernoulli and the $\lambda$-Euler numbers…

An algebraic interpretation of the $q$-Meixner polynomials is obtained. It is based on representations of $\mathcal{U}_q(\mathfrak{su}(1,1))$ on $q$-oscillator states with the polynomials appearing as matrix elements of unitary…

The three canonical families of the hypergeometric orthogonal polynomials (Hermite, Laguerre and Jacobi) control the physical wavefunctions of the bound stationary states of a great deal of quantum systems. The algebraic…

An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is concerned with the behavior of the expected value of $\phi(X)$ for large $\lambda$ where $X$ is a Poisson random variable with mean $\lambda$ and $\phi$ is a…

We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite-Pade approximation to the exponential function, defined by p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are…

In the theory of the Nil-DAHA Fourier transform, the inner products of q-Hermite polynomials for the measure function multiplied by a level one theta function are the key. They are used to obtain expansions of products of any number of such…

We obtain the strong asymptotics of polynomials $p_n(\lambda)$, $\lambda\in\mathbb{C}$, orthogonal with respect to measures in the complex plane of the form $$ e^{-N(|\lambda|^{2s}-t\lambda^s-\overline{t\lambda}^s)}dA(\lambda), $$ where $s$…

We generalize Ramanujan's expansions of the fractional-power Euler functions (q^{1/5})_{\infty} = [ J_1 - q^{1/5} + q^{2/5} J_2 ](q^5)_{\infty} and (q^{1/7})_{\infty} = [ J_1 + q^{1/7} J_2 - q^{2/7} + q^{5/7} J_3 ] (q^7)_{\infty} to…

Using the theory of Calabi-Yau differential equations we obtain all the parameters of Ramanujan-Sato-like series for $1/\pi^2$ as $q$-functions valid in the complex plane. Then we use these q-functions together with a conjecture to find new…

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix-like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the…

Plancherel formula is one of the celebrated result of harmonic analysis on semisimple Lie groups and their homogeneous spaces. The main goal of this work is to find a q-analog of the Plancherel formula for spherical transform the unit…

This article derives full asymptotic expansions for integrals of the form \[ \int_{0}^{1}f(u)(1+q\cdot u^{n})^{w/n}du \] as $n\rightarrow\infty$, with parameters real $w\neq 0$ and $q\in(-1,1]$, or positive $w$ for $q=-1$. We relate the…

We give an explicit solution of a q-Riemann Hilbert problem which arises in the theory of orthogonal polynomials, prove that it is unique, and deduce several properties. Our new results include the asymptotic behaviour of zeroes in the…

We will use a discrete analogue of the classical Laplace method to show that the main term of the asymptotic expansions of certain entire functions, including Ramanujan's entire function $A_{q}(z)$, can be expressed in terms of…

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a…