Note on q-extensions of Euler numbers and polynomials of higher order

In [14] Ozden-Simsek-Cangul constructed generating functions of higher-order twisted $(h,q)$-extension of Euler polynomials and numbers, by using $p$-adic q-deformed fermionic integral on $\Bbb Z_p$. By applying their generating functions, they derived the complete sums of products of the twisted $(h,q)$-extension of Euler polynomials and numbers, see[13, 14]. In this paper we cosider the new $q$-extension of Euler numbers and polynomials to be different which is treated by Ozden-Simsek-Cangul. From our $q$-Euler numbers and polynomials we derive some interesting identities and we construct $q$-Euler zeta functions which interpolate the new $q$-Euler numbers and polynomials at a negative integer. Furthermore we study Barnes' type $q$-Euler zeta functions. Finally we will derive the new formula for " sums products of $q$-Euler numbers and polynomials" by using fermionic $p$-adic $q$-integral on $\Bbb Z_p$.Comment: 11 page

Euler Numbers and polynomials associated with zeta functions

In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.Comment: 12 page

New Results on Higher-Order Daehee and Bernoulli Numbers and Polynomials

We derive new matrix representation for higher order Daehee numbers and polynomials, the higher order lambda-Daehee numbers and polynomials and the twisted lambda-Daehee numbers and polynomials of order k. This helps us to obtain simple and short proofs of many previous results on higher order Daehee numbers and polynomials. Moreover, we obtained recurrence relation, explicit formulas and some new results for these numbers and polynomials. Furthermore, we investigated the relation between these numbers and polynomials and Stirling numbers, Norlund and Bernoulli numbers of higher order. The results of this article gives a generalization of the results derived very recently by El-Desouky and Mustafa [6]