Abstract
In this paper, we consider higher-order Frobenius-Euler polynomials, associated with poly-Bernoulli polynomials, which are derived from polylogarithmic function. These polynomials are called higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials. The purpose of this paper is to give various identities of those polynomials arising from umbral calculus.
Highlights
For λ ∈ C with λ =, the Frobenius-Euler polynomials of order α (α ∈ R) are defined by the generating function to be–λ et – λ α ext = ∞ Hn(α)(x|λ) tn n!. ( . ) n=When x =, Hn(α)(λ) = Hn(α)( |λ) are called the Frobenius-Euler numbers of order α
The Bernoulli polynomials of order α are defined by the generating function to be t et
We recall that the Euler polynomials of order α are given by
Summary
For λ ∈ C with λ = , the Frobenius-Euler polynomials of order α (α ∈ R) are defined by the generating function to be. The Bernoulli polynomials of order α are defined by the generating function to be t et – α ext =. Poly-Bernoulli polynomials are defined by the generating function to be. Let us assume that L|p(x) be the action of the linear functional L on the polynomial p(x), and we remind that the vector space operations on P∗ are defined by L + M|p(x) = L|p(x) + M|p(x) , cL|p(x) = c L|p(x) , where c is a complex constant in C. Let us consider the polynomials Tn(r,k)(x|λ), called higher-order Frobenius-Euler and polyBernoulli mixed-type polynomials, as follows:. When x = , Tn(r,k)(λ) = Tn(r,k)( |λ) is called the nth higher-order Frobenius-Euler and poly-Bernoulli mixed type number.
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