Volume 37 • Number 2 • 2014   • On the Unification of Two Families of Multiple Twisted Type Polynomials by Using $p$-Adic $q$-Integral at $q=-1$ Serkan Araci, Mehmet Acikgoz, Kyoung-Ho Park and Hassan Jolany Abstract. The present paper deals with unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$. Some earlier results of Ozden's papers in terms of unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$ can be deduced. We apply the method of generating function and $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$, which are exploited to derive further classes of Euler polynomials and Genocchi polynomials. To be more precise we summarize our results as follows, we obtain some relations between Ozden's generating function and fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$. Furthermore we derive Witt's type formula for the unification of twisted Euler and Genocchi polynomials. Also we derive distribution formula (Multiplication Theorem) for multiple twisted Euler and Genocchi numbers and polynomials associated with $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$ which yields a deeper insight into the effectiveness of this type of generalizations. Furthermore we define unification of multiple twisted zeta function and we obtain an interpolation formula between unification of multiple twisted zeta function and unification of the multiple twisted Euler and Genocchi numbers at negative integers. Our new generating function possess a number of interesting properties which we state in this paper. 2010 Mathematics Subject Classification: 05A10, 11B65, 28B99, 11B68, 11B73 Full text: PDF