Abstract
In this paper, we give some new and interesting identities which are derived from the basis of Frobenius-Euler. Recently, several authors have studied some identities of Frobenius-Euler polynomials. From the methods of our paper, we can also derive many interesting identities of Frobenius-Euler numbers and polynomials.
Highlights
We develop some new methods to obtain some new identities and properties of Frobenius-Euler polynomials which are derived from the basis of Frobenius-Euler polynomials
Those methods are useful in studying the identities of Frobenius-Euler polynomials
For r ∈ Z+, let us take the rth derivative of g(x) in ( ) as follows: g(r)(x) = ( – λ) k(k – ) · · · (k – r + )bkxk–r, where g(r)(x) dr g (x) dxr
Summary
The Frobienius-Euler polynomials are defined by the generating function to be et. With the usual convention about replacing Hn(x|λ) by Hn(x|λ) (see [ – ]). X = , Hn( |λ) = Hn(λ) are called the nth Frobenius-Euler numbers. H(λ) + n – λHn(λ) = Hn( |λ) – λHn(λ) = ( – λ)δ ,n, where δ ,n is the Kronecker symbol. From ( ), we can derive the following equation: Hn(x|λ) = H(λ) + x n =. By ( ), we see that the leading coefficient of Hn(x|λ) is H (λ) =. Hn(x|λ) are monic polynomials of degree n with coefficients in Q(λ). Hn(x + |λ) – λHn(x|λ) = ( – λ)xn, for n ∈ Z+
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