Some convolution identities for Frobenius-Euler polynomials

Jing Pan,Fengzao Yang

doi:10.1186/s13662-016-1054-5

Jing Pan, Fengzao Yang

Open Access

https://doi.org/10.1186/s13662-016-1054-5

Copy DOI

Abstract

In this paper, by applying the generating function methods and summation transform techniques, we establish some new convolution identities for the Frobenius-Euler polynomials. It turns out that some well-known results are obtained as special cases.

Highlights

The classical Frobenius-Euler polynomials Hn(x|λ) are usually defined by the generating function: et –λ –λ ext = ∞ Hn (x|λ) tn n! n=|t| < π if λ = – ; |t| < log( /λ) otherwise . ( . )In particular, the case x = in ( . ) is called the classical Frobenius-Euler numbers given by Hn(λ) = Hn( |λ)
The case x = in ( . ) is called the classical Frobenius-Euler numbers given by Hn(λ) = Hn( |λ)
It is worthy of mentioning that the classical Frobenius-Euler polynomials and numbers was firstly introduced and studied in great detail by Frobenius [ ]

Summary

Introduction

The classical Frobenius-Euler polynomials Hn(x|λ) are usually defined by the generating function: et. |t| < π if λ = – ; |t| < log( /λ) otherwise. ) is called the classical Frobenius-Euler numbers given by Hn(λ) = Hn( |λ). It is worthy of mentioning that the classical Frobenius-Euler polynomials and numbers was firstly introduced and studied in great detail by Frobenius [ ]. We refer to [ – ] for some interesting properties on the classical Frobenius-Euler polynomials and numbers. The widely investigated analogs of the classical Frobenius-Euler polynomials are the classical Bernoulli polynomials Bn(x) and the classical Euler polynomials En(x), which are usually defined by the generating functions (see, e.g., [ – ]): text tn et – = Bn(x) n!.

The rational numbers Bn and integers En given by

Hm y λ m!

Observe that