Some Identities on the Generalized q-Bernoulli, q-Euler, and q-Genocchi Polynomials

Abstract

Mahmudov (2012, 2013) introduced and investigated some $q$-extensions of the $q$-Bernoulli polynomials ${ℬ}_{n,q}^{\left(\alpha \right)}\left(x,y\right)$ of order $\alpha$, the $q$-Euler polynomials ${ℰ}_{n,q}^{\left(\alpha \right)}\left(x,y\right)$ of order $\alpha$, and the $q$-Genocchi polynomials ${𝒢}_{n,q}^{\left(\alpha \right)}\left(x,y\right)$ of order $\alpha$. In this paper, we give some identities for ${ℬ}_{n,q}^{\left(\alpha \right)}\left(x,y\right)$, ${𝒢}_{n,q}^{\left(\alpha \right)}\left(x,y\right)$, and ${ℰ}_{n,q}^{\left(\alpha \right)}\left(x,y\right)$ and the recurrence relations between these polynomials. This is an analogous result to the $q$-extension of the Srivastava-Pintér addition theorem in Mahmudov (2013).