Higher-order convolutions for Bernoulli and Euler polynomials

Some new results on products of Apostol–Bernoulli and Apostol–Euler polynomials

“…By making use of an elementary idea used by Euler in the discovery of his famous Pentagonal Number Theorem, we establish some new formulae for products of an arbitrary number of Apostol-Bernoulli and ApostolEuler polynomials and numbers. These results are the corresponding generalizations of some known formulae including the higher-order convolution ones discovered by Agoh and Dilcher (2014) [5] on the classical Bernoulli and Euler polynomials. …”

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

“…which is very analogous to the convolution identity on the classical Bernoulli and Euler polynomials presented in [4,Corollaries 1 and 3]. In fact, by using the methods showed in [14], one can derive the similar convolution identity for the classical Genocchi polynomials to Corollary 9 stated in [14].…”

“…We establish some general convolution identities for the Apostol-Bernoulli, ApostolEuler and Apostol-Genocchi polynomials by making use of the generating function methods and summation transform techniques. These results are the corresponding extensions of some known formulas including the general convolution identities on the classical Bernoulli and Euler polynomials due to Dilcher and Vignat [14] and the convolution identities for the classical Genocchi polynomials due to Agoh [4].…”

“…We replace x r by x r − 1 for 1 ≤ r ≤ n in (2.3) to obtain 4) where the product x 1 · · · x r−1 is considered to be equal to 1 when r = 1. If we take x r = λ r e tr for 1 ≤ r ≤ n and substitute k for n in (2.4), then for positive integer k,…”

“…in both sides of the above identity, in view of (1.2) and (2.7), the left hand side of (3.3) can be written as 4) and in light of (1.2) and (2.10), the right hand side of (3.3) can be written as…”