相关论文: Reply to Comment on ``Generating Functions for Her…

Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers…

New methods for derivation of Bell polynomials of the second kind are presented. The methods are based on an ordinary generating function and its composita. The relation between a composita and a Bell polynomial is demonstrated. Main…

We sum multivariate generating functions composed of products of Chebyshev polynomials of the first and the second kind. That is, we find closed forms of expressions of the type $\sum_{j\geq0}\rho^{j}\prod_{m=1}^{k}T_{j+t_{m}}%…

We compute two parametric determinants in which rows and columns are indexed by compositions, where in one determinant the entries are products of binomial coefficients, while in the other the entries are products of powers. These results…

We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the…

A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier-Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical…

We obtain similar types of conclusions as that of Br\"{u}ck [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the…

In this note, we establish some new results on some special types of function algebras and also give new proofs to some existing ones

A new characterization of the generalized Hermite polynomials and of the orthogonal polynomials with respect to the maesure $|x|^\g (1-x^2)^{\a-1/2}dx$ is derived which is based on a "reversing property" of the coefficients in the…

Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we…

The continuous big $q$-Hermite polynomials are shown to realize a basis for a representation space of an extended $q$-oscillator algebra. An expansion formula is algebraically derived using this model.

Based on operator algebras commonly used in quantum mechanics some properties of special functions such as Hermite and Laguerre polynomials and Bessel functions are derived.

We provide an explicit formula for the coefficient polynomials of a Hermite diagonal differential operator. The analysis of the zeros of these coefficient polynomials yields the characterization of generalized Hermite multiplier sequences…

We use analytic combinatorics to give a direct proof of the closed formula for the generating function of $p$-Bernoulli numbers.

Expressions for the derivatives of the Legendre polynomials of the first kind with respect to the order of these polynomials are given. An explicit form for the fourth derivative is presented.

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…

A generalization of Hurwitz stable polynomials to real rational functions is considered. We establishe an analogue of the Hurwitz stability criterion for rational functions and introduce a new type of determinants that can be treated as a…

In this paper, we study umbral calculus to have alternative ways of obtaining our results. That is, we derive some interesting identities of the higher-order Bernoulli, Euler and Hermite polynomials arising from umbral calculus to have…

This addendum presents a relevant stronger consequence of the main theorem of the paper "Higher order stroboscopic averaged functions: a general relationship with Melnikov functions" [arXiv:2011.03663], EJQTDE No. 77 (2021).

In the present paper we generate binary pseudorandom sequences using generalized polynomials. A generalized polynomial is a function in whose description we not only allow addition and product (as it is the case in usual polynomials) but…