Generalized Hermite polynomials and the Bose-like oscillator calculus

Hermite Calculus

We develop a new method of umbral nature to treat blocks of Hermite and of Hermite like polynomials as independent algebraic quantities. The Calculus we propose allows the formulation of a number of practical rules allowing significant simplifications in computational problems.

二元退化 Hermite 多项式及其部分正交性的运算本影方法

Operational-umbral approach to bivariate degenerate Hermite polynomials and their partial orthogonality

本文运用运算微积与本影形式体系研究二元退化 Hermite 多项式，重新推导并扩展其微分方程、单项性、运算恒等式及部分正交性，获得负阶情形的新结果。

New families of q and (q;p)-Hermite polynomials

In this paper, we construct a new family of q-Hermite polynomials denoted by Hn(x,s|q). Main properties and relations are established and proved. In addition, is deduced a sequence of novel polynomials, Ln(. ,.|q), which appear to be connected with well known (q,n)-exponential functions E{q,n}(.), introduced by Ernst in his work entitled: (A New Method for q-calculus, Uppsala Dissertations in Mathematics, Vol. 25, 2002). Relevant results spread in the literature are retrieved as particular cases. Fourier integral transforms are explicitly computed and discussed. A (q;p)-extension of the Hn(x,s…

Continuous and discrete fractional operators and some fractional functions

The classical orthogonal polynomials are usually defined by the Rodrigues' formula. This paper refers to a fractional extension of the classical Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk and Hahn polynomials. By means of the Caputo operator of fractional calculus, C-Hermite, C-Laguerre, C-Legndre and the C-Jacobi functions are defined and their representation in terms of the hypergeometric functions are provided. Also, by means of the Gray and Zhang fractional difference oparator, fractional Charlier, Meixner, Krawtchouk and Hahn functions are defined and their representation in…

Operational Methods in the Study of Sobolev-Jacobi Polynomials

Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover…