Notes on Unified Q-Apostol-Type Polynomials

Burak Kurt

Recently, many mathematicians (Karande and Thakare [6], Ozarslan [14], Ozden et. al. [15], El-Deouky et. al. [5]) have studied the unification of Bernoulli, Euler and Genocchi polynomials. They gave some recurrence relations and proved some theorems. Mahmudov [13] defined the new $q$-Apostol--Bernoulli and $q$-Apostol--Euler polynomials. Also he gave the analogous of the Srivastava--Pintér addition theorems. Kurt [8] 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 we derive many new identities and recurrence relations associated with the $q$-Apostol-type Bernoulli, the $q$-Apostol-type Euler and the $q$-Apostol-type Genocchi polynomials and numbers and also the generalized Stirling type numbers of the second kind.