Abstract
The main purpose of this paper is, using the generating function methods and summation transform techniques, to establish some new formulas for the products of an arbitrary number of the Frobenius-Euler polynomials and give some illustrative special cases.
Highlights
Let λ be a complex number with λ =
Frobenius [ ] introduced and studied the so-called Frobenius-Euler polynomials Hn(x|λ), which are usually defined by the following exponential generating function:
En = nEn are called the Bernoulli numbers and the Euler numbers, respectively. It is seen from ( . ) and ( . ) that the Frobenius-Euler polynomials give the Euler polynomials when λ = – in ( . ), and the Bernoulli polynomials can be expressed by the Frobenius-Euler polynomials as follows: m
Summary
) gives the Frobenius-Euler numbers Hn(λ) = Hn( |λ). Where, and in what follows, a k is the binomial coefficient defined for a complex number a and a non-negative integer k by a = , a a(a – )(a – ) · · · (a – k + ) = ), and the Bernoulli polynomials can be expressed by the Frobenius-Euler polynomials as follows: m–
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