Abstract
In the present paper, we find the Fourier expansion of the Apostol Frobenius–Euler polynomials. By using a Fourier expansion of the Apostol Frobenius–Euler polynomials, we derive some new and interesting results.
Highlights
We begin some known definitions and properties of Frobenius–Euler polynomials and Apostol Frobenius–Euler polynomials which will be useful in deriving the main results of this paper
When Fourier was trying to solve a problem in heat conduction, he needed to express a function f as an infinite series of sine and cosine functions: f (x) = a0 + 2 ∞(an cos nx + bn sin nx). n=1Earlier, Bernoulli and Euler had used such series while investigating problems concerning vibrating strings and astronomy
5 Conclusion and observation In the paper, we have derived the Fourier expansion of Apostol Frobenius–Euler polynomials as Theorem 1
Summary
We begin some known definitions and properties of Frobenius–Euler polynomials and Apostol Frobenius–Euler polynomials which will be useful in deriving the main results of this paper. Definition 3 The Frobenius–Genocchi polynomials are defined by means of the following generating function: Main results We begin with the following theorem, which is a Fourier series expansion of the Apostol Frobenius–Euler polynomial. The first proof includes the Cauchy residue theorem and a complex integral over a circle C following Bayad’s method in [4].
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