arXiv subjects
Publications and source records attributed to Ruiming Zhang.
In this note, we present a systematic method to explicitly compute the determinants and inverses for some generalized Hilbert matrices associated with orthogonal systems with explicit representations. We expressed the determinant, the inverse and a lower bound for the smallest eigenvalue of such matrix in terms of the explicit orthogonal system.
In this note, we demonstrate a method to invert some Hankel matrices explicitly by using the kernel polynomials for the related classical orthogonal polynomials.
In this note, we apply kernel polynomials to find the explicit inverses for some some Hankel matrices associated with q-orthogonal polynomials.
Key words and phrases: q-Airy function (Ramanujan's entire function); q-Bessel function; Bessel function; Airy function; Riemann zeta function; Dirichlet L-series.
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.
In this paper we derive some asymptotic formulas for the $q$-Gamma function $Γ_{q}(z)$ for $q$ tending to 1.
In this work we investigate the asymptotics for Euler's $q$-Exponential $E_{q}(z)$, $q$-Gamma function $Γ_{q}(z)$, Ramanujan's function $A_{q}(z)$, Jackson's $q$-Bessel function $J_ν^{(2)}$(z;q) of second kind, Stieltjes-Wigert orthogonal polynomials $S_{n}(x;q)$ and $q$-Laguerre polynomials $L_{n}^{(α)}(x;q)$ as $q$ approaching 1.
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 complex plane, its number theoretical property completely determines the order of the error term. In each cases, we also investigate the asymptotic behavior of the mentioned $q$-series when $q$ approaching 1 in a restricted manner. These asymptotic formulas may provide insights to new random matrix models.
In this work we study the Plancherel-Rotach type asymptotics for Stieltjes-Wigert orthogonal polynomials with complex scaling. The main term of the asymptotics contains Ramanujan function $A_{q}(z)$ for the scaling parameter on the vertical line $\Re(s)=2$, while the main term of the asymptotics involves the theta functions for the scaling parameter in the strip $0<\Re(s)<2$. In the latter case the number theoretical property of the scaling parameter completely determines the order of the error term. $ $ These asymptotic formulas may provide some insights to some new random matrix model and also add a new link between special functions and number theory.
In this work we study the Plancherel-Rotach type asymptotics for $q$-Laguerre orthogonal polynomials with complex scaling . The main term of the asymptotics contains Ramanujan function $A_{q}(z)$ for the scaling parameter on the vertical line $\Re(s)=2$, while the main term of the asymptotics involves the theta function for the scaling parameter in the strip $0<\Re(s)<2$. In the latter case the number theoretical property of the scaling parameter completely determines the order of the error term. $ $These asymptotic formulas may provide insights to some new random matrix model and add a new link between special functions and number theory.
In this work we study the Plancherel-Rotach type asymptotics for Ismail-Masson orthogonal polynomials with complex scaling. The main term of the asymptotics contains Ramanujan function $A_{q}(z)$ for the scaling parameter on the vertical line $\Re(s)={1/2}$, while the main term of the asymptotics involves the theta functions for the scaling parameter in the strip $0<\Re(s)<{1/2}$. In the latter case the number theoretical property of the scaling parameter completely determines the order of the error term. $ $These asymptotic formulas may provide insights to some new random matrix models and also add a new link between special functions and number theory.
In this short notes we will derive an inequality for scaled $q^{-1}$-Hermite orthogonal polynomials of Ismail and Masson, an inequality for scaled Stieltjes-Wigert, two inequalities for Ramanujan function and two definite integrals for Ramanujan function.
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 $θ$-functions.
In this very short note we will derive an inequality for a class of entire functions including all the confluent basic hypergeometric series and an inequality for a class of meromorphic functions including theta functions.
We will use a discrete analogue of the classical \emph{Laplace} method to show that for infinitely many positive integers $n$, the main term of the asymptotic expansions of scaled confluent basic hypergeometric functions, including the Ramanujan's entire function $A_{q}(z)$, could be expressed in $θ$-functions, and the error term depends on the ergodic property of certain real scaling parameter.
We will use a discrete analogue of the classical Laplace method to show that for infinitely many positive integers $n$, the main term of the asymptotic expansion of the scaled $q$-exponential $(-q^{-nt+1/2}u;q)_{\infty}$ could be expressed in $θ$-function.
We establish an integral representations of a right inverses of the Askey-Wilson finite difference operator in an $L^2$ space weighted by the weight function of the continuous $q$-Jacobi polynomials. We characterize the eigenvalues of this integral operator and prove a $q$-analog of the expansion of $e^{ixy}$ in Jacobi polynomials of argument $x$. We also outline a general procedure of finding integral representations for inverses of linear operators.