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In 2009, Kim et al. gave some identities of symmetry for the twisted Euler polynomials of higher-order, recently. In this paper, we extend our result to the higher-order twisted -Euler numbers and polynomials. The purpose of this paper is to establish various identities concerning higher-order twisted -Euler numbers and polynomials by the properties of -adic invariant integral on . Especially, if , we derive the result of Kim et al. (2009).

This paper performs a further investigation on the q-Bernoulli polynomials and numbers given by Acikgoz et al. (Adv. Differ. Equ. 2010, 9, Article ID 951764) some incorrect properties are revised. It is pointed out that the definition concerning the q-Bernoulli polynomials and numbers is unreasonable. The purpose of this paper is to redefine the q-Bernoulli polynomials and numbers and correct its wrong properties and rebuild its theorems.

Recently, many mathematicians have studied various kinds of the -analogue of Genocchi numbers and polynomials. In the work (New approach to q-Euler, Genocchi numbers and their interpolation functions, "Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 105–112, 2009.", Kim defined new generating functions of -Genocchi, -Euler polynomials, and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz type -zeta function. This function interpolates -Genocchi polynomials at negative integers. Finally, we also give some identities related to these polynomials.

The main purpose of this paper is to study the distribution of Genocchi polynomials. Finally, we construct the Genocchi zeta function which interpolates Genocchi polynomials at negative integers.

Recently, many authors have studied twisted -Bernoulli polynomials by using the -adic invariant -integral on . In this paper, we define the twisted -adic -integral on and extend our result to the twisted -Bernoulli polynomials and numbers. Finally, we derive some various identities related to the twisted -Bernoulli polynomials.