Notes on Unified q-Apostol-Type Polynomials

Abstract

Recently, many mathematicians (Karande and Thakare, Ozarslan, Ozden et. al., El-Deouky et. al.) have studied the unification of Bernoulli, Euler and Genocchi polynomials. They gave some recurrence relations and proved some theorems. Mahmudov defined the new q-Apostol-Bernoulli and q-Apostol-Euler polynomials. Also he gave the analogous of the Srivastava-Pintér addition theorems. Kurt  gave the new identities and some relations for these polynomials.
In this work, we give some recurrence relations for the unified q-Apostol-type polynomials related to multiple sums. By using generating functions for the unification of the q-Apostol-type Bernoulli, Euler and Genocchi polynomials and numbers, we derive many new identities and recurrence relations for these polynomials. We obtain a new identity related to the generalized Stirling type numbers of the second kind.