Abstract
In this paper, a further investigation for the Apostol-Bernoulli and Apostol-Euler polynomials and numbers is performed. Some closed formulae of sums of products of any number of Apostol-Bernoulli and Apostol-Euler polynomials and numbers are established by applying the generating function methods and some summation transform techniques. It turns out that some well-known results are derived as special cases. MSC:11B68, 05A19.
Highlights
The classical Bernoulli polynomials Bn(x) and Euler polynomials En(x) are usually defined by means of the following generating functions: text ∞tn et – = Bn(x) n! |t| < π ext ∞tn and et + = En(x) n! |t| < π . ( . ) n=In particular, the rational numbers Bn = Bn( ) and integers En = nEn( / ) are called the classical Bernoulli numbers and Euler numbers, respectively.As is well known, the classical Bernoulli and Euler polynomials and numbers play important roles in different areas of mathematics such as number theory, combinatorics, special functions and analysis
Kim [ ] developed a new approach to give the closed formula of sums of products of any number of classical Bernoulli numbers by using the relation of values at non-positive integers of the important representation of the multiple Hurwitz zeta function in terms of the Hurwitz zeta function
We explore the closed formulae of sums of products of any number of ApostolBernoulli and Apostol-Euler polynomials and numbers
Summary
Dilcher [ ] obtained some closed formulae of sums of products of any number of classical Bernoulli and Euler polynomials and numbers. Kim [ ] developed a new approach to give the closed formula of sums of products of any number of classical Bernoulli numbers by using the relation of values at non-positive integers of the important representation of the multiple Hurwitz zeta function in terms of the Hurwitz zeta function. Kim and Hu [ ] obtained the closed formula of sums of products of any number of ApostolBernoulli numbers by expressing the sums of products of the Apostol-Bernoulli polynomials in terms of the special values of multiple Hurwitz-Lerch zeta functions at non-positive integers.
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