Journal of the
Korean Mathematical Society JKMS

The main purpose of this paper is to investigate the $q$-Apostol type Frobenius-Euler numbers and polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive $q$-integers. By using infinite series representation for $q$-Apostol type Frobenius-Euler numbers and polynomials including their interpolation functions, we not only give some identities and relations for these numbers and polynomials, but also define generating functions for new numbers and polynomials. Further we give remarks and observations on generating functions for these new numbers and polynomials. By using these generating functions, we derive recurrence relations and finite sums related to these numbers and polynomials. Moreover, by applying higher-order derivative to these generating functions, we derive some new formulas including the Hurwitz--Lerch zeta function, the Apostol-Bernoulli numbers and the Apostol-Euler numbers. Finally, for an application of the generating functions, we derive a multiplication formula, which is very important property in the theories of normalized polynomials and Dedekind type sums.

Keywords: generating functions, special numbers and polynomials, Euler numbers, Frobenius-Euler numbers, Apostol-Bernoulli numbers, Apostol-Euler numbers, $q$-Apostol type Frobenius-Euler numbers and polynomials, $l_{q}$-series, multiplication formula

MSC numbers: 05A10, 05A15, 05A30, 11B37, 11B68, 11B83, 26C05, 34A99, 35A99

Supported by: The present investigation was supported by the Scientific Research Project Administration of the University of Akdeniz under Project Number FBA-2018-3292.