Euler Matrices and their Algebraic Properties Revisited (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.

The Representations of the Fibonacci and Lucas Matrices

Iranian Journal of Science and Technology, Transactions A: Science, 2019

In this study, a matrix R L is defined by the properties associated with the Pascal matrix, and two closed-form expressions of the matrix function f (R L) = R n L are determined by methods in matrix theory. These expressions satisfy a connection between the integer sequences of the first-second kinds and the Pascal matrices. The matrix R n L is the Fibonacci Lucas matrix, whose entries are the Fibonacci and Lucas numbers. Also, the representations of the Lucas matrix are derived by the matrix function f (R L − 5I) , and various forms of the matrix (R L − 5I) n in terms of a binomial coefficient are studied by methods in number theory. These representations give varied ways to obtain the new Fibonacci-and Lucas-type identities via several properties of the matrices R n L and (R L − 5I) n .

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.

On Some Generalizations of the Vandermonde Matrix and Their Relations with the Euler Beta-Function

Georgian Mathematical Journal, 1994

A multiple Vandermonde matrix which, besides the powers of variables, also contains their derivatives is introduced and an explicit expression of its determinant is obtained. For the case of arbitrary real powers, when the variables are positive, it is proved that such generalized multiple Vandermonde matrix is positive definite for appropriate enumerations of rows and columns. As an application of these results, some relations are obtained which in the one-dimensional case give the well-known formula for the Euler betafunction.

New Generalization of Eulerian polynomials and Their Application

J. Ana. Num. Theor. 2, No. 2, 59-63 (2014)

In the present paper, we introduce Eulerian polynomials with a and b parameters and give the definition of them. By using the definition of generating function for our polynomials, we derive some new identities in Theory of Analytic Numbers. Also, we give relations between Eulerian polynomials with a and b parameters, Bernstein polynomials, Poly-logarithm function, Bernoulli numbers and Euler numbers. Moreover, we see that our polynomials at a = 1 are related to Euler-Zeta function at negative inetegers which we express in this paper.

New Generalization of Eulerian polynomials and their applications

arXiv preprint arXiv:1208.1271, 2012

Abstract: In the present paper, we introduce Eulerian polynomials with a and b parameters and give the definition of them. By using the definition of generating function for our polynomials, we derive some new identities in Theory of Analytic Numbers. Also, we give relations between Eulerian polynomials with a and b parameters, Bernstein polynomials, Poly-logarithm function, Bernoulli numbers and Euler numbers. Moreover, we see that our polynomials at a=-1 are related to Euler-Zeta function at negative inetegers. Finally, we ...