The modified q-Genocchi numbers and polynomials with applications to q-zeta functions

In this paper, we introduce the modified q-Genocchi polynomials, investigate their properties, and give their generating function. We obtain a number of new relations and properties for q-Genocchi polynomials, such as addition formula, explicit formula, derivative formula, and multiplication formula. As an application, a new q-analogue of Genocchi zeta function is presented by applying the Mellin transform to the generating function of the modified q-Genocchi polynomials. Finally, we define the q-Genocchi zeta-type functions and then prove their interpolation by the modified q-Genocchi polynomials at negative integers.