General convolution identities for Apostol-Bernoulli, Euler and Genocchi polynomials

Abstract

We perform a further investigation for the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. By making use of the generating function methods and summation transform techniques, we establish some general convolution identities for the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. These results are the corresponding extensions of some known formulas including the general convolution identities discovered by Dilcher and Vignat [K.

Parametric kinds of generalized Apostol--Bernoulli polynomials and their properties

“…In the literature, there are many studies on the Apostol-Bernoulli polynomials such as Gaussian hypergeometric function, multiplication formulas, and fourier expansions. The relations between the Apostol-Bernoulli polynomials, the Euler polynomials, the Genocchi polynomials, and the Stirling numbers have also been investigated in [12,13,17,18,21,22,41].…”

Higher-Order Convolutions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials

“…We also denote by N and N * the set of positive integers and the set of non-negative integers, respectively. For α, λ ∈ C, the generalized Apostol-Bernoulli polynomials B (α) n (x; λ), the generalized Apostol-Euler polynomials E (α) n (x; λ) and the generalized Apostol-Genocchi polynomials G (α) n (x; λ) of order α are defined by the following generating functions (see, e.g., [1][2][3][4]):…”

On the Lucas polynomials and some of their new identities

“…Wang and Zhang [4] proved some divisible properties involving Fibonacci numbers and Lucas numbers. Some of other related papers can also be found in references [5][6][7][8][9][10][11][12][13][14][15], here we are not going to list them all.…”