Related papers: Mehler's formulas for the univariate complex Hermi…
The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to…
We show that sampling or interpolation formulas in reproducing kernel Hilbert spaces can be obtained by reproducing kernels whose dual systems form molecules, ensuring that the size profile of a function is fully reflected by the size…
In this paper, using the technical tools in \cite{TW5}, we solve the complex Hessian equation on closed Hermitian manifolds, which generalizes the the K\"ahler case results in \cite{HMW} and \cite{DK}.
In this paper, we define multi poly-Bernoulli polynomials using multiple polylogarithm and derive some properties parallel to those of poly-Bernoulli polynomials. Furthermore, an explicit formula for certain Hurwitz-Lerch type multi…
We show that the formalism of hybrid polynomials, interpolating between Hermite and Laguerre polynomials, is very useful in the study of Motzkin numbers and central trinomial coefficients. These sequences are identified as special values of…
The multivariable version of ordinary and generalized Hermite polynomials are the natural solutions of the classical heat equation and of its higher order versions. We derive the associated Burgers equations and show that analogous…
In this paper, we resurrect a long-forgotten notion of equivalence for univariate polynomials with integral coefficients introduced by Hermite in the 1850s. We show that the Hermite equivalence class of a polynomial has a very natural…
We introduce a new class of holomorphic polynomials extending the classical Gould--Hopper to two complex variables. The considered polynomials include the $1$-D and $2$-D holomorphic and polyanalytic It\^o--Hermite polynomials as particular…
In this paper, we investigate the $\partial$-complex on weighted Bergman spaces on Hermitian manifolds satisfying a certain holomorphicity/duality condition. This generalizes the situation of the Segal-Bargmann space in $\mathbb{C}^n$,…
We study the orthogonal complement of the Hilbert subspace considered by by van Eijndhoven and Meyers in [J. Math. Anal. Appl. 146 (1990), no. 1, 89--98} and associated to holomorphic Hermite polynomials. A polyanalytic orthonormal basis is…
The operational calculus associated with Hermite numbers has been shown to be an effective tool for simplifying the study of special functions. Within this context, Hermite polynomials have been viewed as Newton binomials, with the…
In this paper, we develop the theory of weighted Bergman space and obtain a general representation formula of the Bergman kernel function for the spaces on the Reinhardt domain containing the origin. As applications, we calculate the…
We prove that for |x|,|t|<1, -1 <q \leq1 and n\geq0: \Sigma_{i\geq0}((t^{i})/((q)_{i}))h_{n+i}(x|q) = h_{n}(x|t,q) \Sigma_{i\geq0}((t^{i})/((q)_{i}))h_{i}(x|q), where h_{n}(x|q) and h_{n}(x|t,q) are respectively the so called q-Hermite and…
We use generating functions to express orthogonality relations in the form of $q$-beta integrals. The integrand of such a $q$-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal
We present two types of polynomials related to the Mittag-Leffler function namely the fractional Hermite polynomial and the Mittag-Leffler polynomial. The first modifies the Hermite polynomial and the second one is a refashioned Laguerre…
It is shown that fundamental solutions $K^\sigma(x,y;t)=\langle x|e^{-i H^\sigma t}|y\rangle$ of the non-stationary Schr\"{o}dinger equation (Green functions, or propagators) for the rational extensions of the Harmonic oscillator…
In this paper, we continue our study of the Hilbert polynomials of coinvariants begun in our previous work math.QA/0205324 (paper I). We describe the sl_n-fusion products for symmetric tensor representations following the method of Feigin…
We establish an operator--theoretic correspondence between periodic Bernoulli kernels and Hermite polynomials, framed through the umbral calculus and a quantum analogy. Starting from the analytic master function $F^\ast$, the periodic…
By applying Slater's transformation formulas for the bilateral basic hypergeometric series ${}_2\psi_{2}$, we derive three type translation formulas for the generalized Zwegers' $\mu$-function (``continuous $q$-Hermite function'') which was…
In this work we show how to get advantage from the Riemann--Hilbert analysis in order to obtain information about the matrix orthogonal polynomials and functions of second kind associated with a weight matrix. We deduce properties for the…