Abstract
In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. Some results presented here are the corresponding extensions of several known formulas.
Highlights
Throughout this paper, C and C× denote the set of complex numbers and the set of complex numbers excluding zero, respectively
Polynomials Gn ( x; λ) of order α are defined by the following generating functions: t λet − 1
We denote by s(n, k) the Stirling numbers of the first kind and by S(n, k ) the Stirling numbers of the second kind, which are usually defined by the following generating functions: ln(1 + t) s(n, k) n=k tn n!
Summary
Throughout this paper, C and C× denote the set of complex numbers and the set of complex numbers excluding zero, respectively. Bn ( x; λ), the generalized Apostol-Euler polynomials En ( x; λ) and the generalized Apostol-Genocchi polynomials Gn ( x; λ) of order α are defined by the following generating functions (see, e.g., [1,2,3,4]):. Apostol-Euler polynomials and the generalized Apostol-Genocchi polynomials was unified by the following generating function (see, for example, [5]): 21 − κ t κ βb et − ab. We shall be concerned with some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. We establish several higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials by making use of the generating-function methods and summation-transform techniques.
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