Matrix approach to Frobenius-Euler polynomials (original) (raw)

On F-Frobenius-Euler polynomials and their matrix approach

Journal of Mathematics and Computer Science, 2023

In this article, the generalized F-Frobenius-Euler polynomials H (α) n,F (x; µ) are introduced, through their generating function, and properties are established for these generalized polynomials. In addition, we define the generalized polynomial Fibo-Frobenius-Euler matrix H (α) n (x, F, µ). Factorizations of the Fibo-Frobenius-Euler polynomial matrix are established with the generalized Fibo-Pascal matrix and the Fibonacci matrix. The inverse of the Fibo-Frobenius-Euler matrix is also found.

New Results Parametric Apostol-Type Frobenius-Euler Polynomials and their Matrix Approach

Kragujevac Journal of Mathematics, 2024

The new algebraic properties of the parametric Apostol-type Frobenius-Euler polynomials and parametric type Frobenius-Euler polynomial have been explained in this research. The researchers have studied the series of the Taylor type and established the relation between the classic Apostol Frobenius-Euler and Frobenius-Euler polynomials. This work has also addressed the generalized parametric Apostol-type Frobenius-Euler polynomials matrices and has shown some of their properties. Finally, this research provided some factorizations of Apostol-type Frobenius-Euler matrix that involves the generalized Pascal matrix, Fibonacci and Lucas matrices, respectively.

NEW GENERALIZED APOSTOL-FROBENIUS-EULER POLYNOMIALS AND THEIR MATRIX APPROACH

In this paper, we introduce a new extension of the generalized Apostol-Frobenius-Euler polynomials H [m−1,α] n (x; c, a; λ; u). We give some algebraic and differential properties, as well as, relationships between this polynomials class with other polynomials and numbers. We also, introduce the generalized Apostol-Frobenius-Euler polynomials matrix U [m−1,α] (x; c, a; λ; u) and the new generalized Apostol-Frobenius-Euler matrix U [m−1,α] (c, a; λ; u), we deduce a product formula for U [m−1,α] (x; c, a; λ; u) and provide some factorizations of the Apostol-Frobenius-Euler polynomial matrix U [m−1,α] (x; c, a; λ; u), which involving the generalized Pascal matrix.

Some explicit formulas for the generalized Frobenius-Euler polynomials of higher order

Filomat, 2019

Our aim is to derive some explicit formulas for the generalized Bernoulli and Euler polynomials in terms of Whitney and translated Whitney numbers of the second kind. Also we derive some explicit formulas for the generalized Euler polynomials and Genocchi-like polynomials in terms of generalized Whitney polynomials of the second kind. We provide an algorithm for computing the generalized Frobenius-Euler polynomials of higher order.

On the generalized Apostol-type Frobenius-Euler polynomials

Advances in Difference Equations, 2013

The aim of this paper is to derive some new identities related to the Frobenius-Euler polynomials. We also give relation between the generalized Frobenius-Euler polynomials and the generalized Hurwitz-Lerch zeta function at negative integers. Furthermore, our results give generalized Carliz's results which are associated with Frobenius-Euler polynomials.

A unified matrix approach to the representation of Appell polynomials

Integral Transforms and Special Functions, 2015

In this paper we propose a unified approach to matrix representations of different types of Appell polynomials. This approach is based on the creation matrix -a special matrix which has only the natural numbers as entries and is closely related to the well known Pascal matrix. By this means we stress the arithmetical origins of Appell polynomials. The approach also allows to derive, in a simplified way, the properties of Appell polynomials by using only matrix operations.

On the generalized Euler-Frobenius polynomial

Journal of Approximation Theory, 1981

In this paper, the properties of the generalized Euler-Frobenius polynomial IZ,(., 9) are studied. It is proved that its zeroes are separated by the factor q and their asymptotic behavior, as q + co, is given. As a consequence, it is shown that least squares spline approximation on a biinfinite geometric mesh can be bounded independently of the (local) mesh ratio q and that the norm of the inverse of the corresponding order k B-spline Gram matrix decreases monotonically to 2k-1 for large q. as q+ m.

Truncated-exponential-based Frobenius–Euler polynomials

Advances in Difference Equations, 2019

In this paper, we first introduce a new family of polynomials, which are called the truncated-exponential based Frobenius–Euler polynomials, based upon an exponential generating function. By making use of this exponential generating function, we obtain their several new properties and explicit summation formulas. Finally, we consider the truncated-exponential based Apostol-type Frobenius–Euler polynomials and their quasi-monomial properties.

A New Class of Hermite-Apostol Type Frobenius-Euler Polynomials and Its Applications

Symmetry, 2018

The article is written with the objectives to introduce a multi-variable hybrid class, namely the Hermite–Apostol-type Frobenius–Euler polynomials, and to characterize their properties via different generating function techniques. Several explicit relations involving Hurwitz–Lerch Zeta functions and some summation formulae related to these polynomials are derived. Further, we establish certain symmetry identities involving generalized power sums and Hurwitz–Lerch Zeta functions. An operational view for these polynomials is presented, and corresponding applications are given. The illustrative special cases are also mentioned along with their generating equations.