Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions.(English)Zbl 1173.11064

The author continues his long series of papers around Bernoulli numbers and polynomials, Riemann’s and Hurwitz’s zeta function and Dirichlet’s $$L$$-functions, in particular their twisted and $$q$$-twisted versions. (Here $$q$$ denotes either a complex or a $$p$$-adic number satisfying certain conditions.) The main purpose of the present article is, as the author says in the abstract, to study generating functions of twisted and $$q$$-twisted Bernoulli numbers and polynomials. The functions mentioned above are constructed from these by using the Mellin transform. The article seems to be partly of a survey type with numerous references to previous results, and it is not always clear what is new.
There are many lapses and some confusing statements showing that the author should have checked his text more carefully.

MSC:

 11S40 Zeta functions and $$L$$-functions 11B68 Bernoulli and Euler numbers and polynomials 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 44A20 Integral transforms of special functions