Matrix Approach to Frobenius-Euler Polynomials

Abstract

Abstract. In the last two years Frobenius-Euler polynomials have gained renewed interest and were studied by several authors. This paper presents a novel approach to these polynomials by treating them as Appell polynomials. This allows to apply an elementary matrix representation based on a nilpotent creation matrix for proving some of the main properties of Frobenius-Euler polynomials in a straightforward way.

Generalized Fubini transform with two variables

“…The general term of the Frobenius-Euler polynomial sequence (H(x; y)) n≥0 defined by the generating function [11] (4.20)…”

Asymptotic Approximations of Higher-Order Apostol–Frobenius–Genocchi Polynomials with Enlarged Region of Validity

“…The Frobenius-Genocchi polynomials are formed by mixing the definitions of the two classical polynomials, namely the Frobenius polynomials and the Genocchi polynomials. The Frobenius polynomials and numbers can be traced back to the works of the great German mathematician Ferdinand Georg Frobenius, who studied the context of these polynomials in number theory and the relation of its divisibility properties with the Stirling numbers of the second kind [6]. On the other hand, the Genocchi polynomials, which were named after Angelo Genocchi, have been studied extensively because of their relevant combinatorial relations and properties in number theory, complex analytic number theory, homotopy theory, quantum physics, etc.…”

“…, which are given by w j = 2jπ − δ, j = ±1, ±2, • • • . By taking z → nz and letting nz → ∞ with fixed z in (6),…”

Generalized Fubini transform with two variables

“…Now, in this part of the paper, we will connect the polynomials F n (x, y) with Eulerian polynomials and Frobenius-Euler polynomials. It is known that for x ̸ = 1 and n ≥ 0, the Eulerian polynomials A n (x) and the Frobenius-Euler polynomials H n (x; y) are defined respectively by the following generating functions [12,13] 1…”