Related papers: Generalized Zernike polynomials: Operational formu…
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…
The Brenke type generating functions are the polynomial generating functions of the form $$\sum_{n=0}^{\infty}{P_n(x )\over n!}t^n=A(t)B(xt), $$ where $A$ and $B$ are two formal power series subject to the conditions…
In this paper we generalize and specialize generating functions for classical orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
Our goal in this work is to found a closed form for rational generat- ing functions, these generate a various families of polynomials and generalized polynomials, in order to get the general recursive formula satisfied by these polynomials.
We introduce a class of doubly indexed real Hermite polynomials and we deal with their related properties like the associated recurrence formulae, Runge's addition formula, generating function and Nielsen's identity.
The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study…
A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained,…
The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions.…
The radial polynomials of the 2D (circular) and 3D (spherical) Zernike functions are tabulated as powers of the radial distance. The reciprocal tabulation of powers of the radial distance in series of radial polynomials is also given, based…
In this paper, we define new generalized k-Mersenne numbers and give a formula of generalized Mersenne polynomials and further we study their properties. Moreover, we define Gaussian Mersenne numbers and obtain some identities like Binet…
We realize that geometric polynomials and p-Bernoulli polynomials and numbers are closely related with an integral representation. Therefore, using geometric polynomials, we extend some properties of Bernoulli polynomials and numbers such…
In this paper, we derive some explicit expansion formulas associated to Brenke polynomials using operational rules based on their corresponding generating functions. The obtained coefficients are expressed either in terms of finite double…
We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear…
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…
The relations between the Bernoulli and Eulerian polynomials of higher order and the complete Bell polynomials are found that lead to new identities for the Bernoulli and Eulerian polynomials and numbers of higher order. General form of…
Studying degenerate versions of various special polynomials have become an active area of research and yielded many interesting arithmetic and combinatorial results. Here we introduce a degenerate version of polylogarithm function, called…
The main purpose and motivation of this article is to create a linear transformation on the polynomial ring of rational numbers. A matrix representation of this linear transformation based on standard fundamentals will be given. For some…
The use of operational methods of different nature is shown to be a fairly powerful tool to study different problems regarding the theory of Legendre and Legendre-like polynomials. We show how the use of the well known integral…
A generalization of the generating function for Gegenbauer polynomials is introduced whose coefficients are given in terms of associated Legendre functions of the second kind. We discuss how our expansion represents a generalization of…