Related papers: Some classes of generating functions for generaliz…
In a previous paper, Rahmani introduced a new family of p-Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. Motivated by this paper and as a degenerate version of those numbers and polynomials, we introduce…
We give two widest Mehler's formulas for the univariate complex Hermite polynomials $H_{m,n}^\nu$, by performing double summations involving the products $u^m H_{m,n}^\nu (z,\overline{z}) \overline{H_{m,n}^\nu (w,\overline{w})}$ and $u^m…
The aim of this paper is twofold. Firstly, we investigate a finite sum involving the generalized falling factorial polynomials, in some special cases of which we express it in terms of the degenerate Stirling numbers of the second kind, the…
This article aims to reinforce the broad applicability of the umbral approach to address complex mathematical challenges and contribute to various scientific and engineering endeavors. The umbral methods are used to reformulate the…
We show that the use of generalized multivariable forms of Hermite polynomials provide an useful tool for the evaluation of families of elliptic type integrals often encountered in electrostatic and electrodynamics
We obtain a series transformation formula involving the classical Hermite polynomials. We then provide a number of applications using appropriate binomial transformations. Several of the new series involve Hermite polynomials and harmonic…
We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating…
We give a new construction of q-Genocchi numbers, Euler numbers of higher order, which are different than the q-Genocchi numbers of Cangul-Ozden-Simsek. By using our q-Genoucchi, Euler nimbers of higher order, we can investigate the…
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…
Recently subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called P\'olya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the…
We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function and concepts of the Umbral Calculus associated with it. Also, we present…
According to generalized Mellin derivative (Kargin), we introduce a new family of polynomials called higher order generalized geometric polynomials. We obtain some properties of them.We discuss their connections to degenerate Bernoulli and…
We give new explicit representations as well as new generating functions for the associated Meixner, Charlier, Laguerre, and Krawtchouk polynomials. The obtained results are then used to derive new generating functions and convolution-type…
Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…
Inspired by the framework of operational methods and based on the generating functions of Legendre-Gould Hopper polynomials and Sheffer sequences, we discuss certain new mixed type polynomials and their important properties. We show that…
The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these…
The exponential generating function of ordinary generating functions of diagonal sequences of general Sheffer triangles is computed by an application of Lagrange's theorem. For the special Jabotinsky type this is already known. An analogous…
We introduce a new array of type $D$ Eulerian numbers, different from that studied by Brenti, Chow and Hyatt. We find in particular the recurrence relation, Worpitzky formula and the generating function. We also find the probability…
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…
In this paper, we introduce a certain method to construct polynomials producing many absolute pseudoprimes. By this method, we give new polynomials producing absolute pseudoprimes with any fixed number of prime factors which can be viewed…