On p-adic interpolating function associated with modified Dirichlet's type of twisted q-Euler numbers and polynomials with weight α

Abstract

Abstract. In the present paper, we introduce modified Dirichlet's type of twisted q -Euler polynomials with weight α . We apply the method of generating function and p -adic q -integral representation on Z p , which are exploited to derive further classes of q -Euler numbers and polynomials. Our new generating function possess a number of interesting properties which we state in this paper.

On the Properties of q-Bernstein-type Polynomials

“…The values of the negative integer points, also found by Euler, are rational numbers and play a vital and important role in the theory of modular forms. Many generalization of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known in [18], [24], [25], [26], [9], [10]. So, we construct interpolation function of modified q-Bernstein polynomials of several variables.…”

“…If q ∈ C p we normally assume that |q − 1| p < p − 1 p−1 so that q x = exp (x log q) for |x| p ≤ 1 cf. [4], [5], [7], [8], [9], [10], [31], [13], [14]. Let UD (Z p ) be the set of uniformly differentiable function.…”

“…The Euler numbers are defined by E n (1/2) = 2 n E n . The Euler polynomials can be generated through Equation (for details, see [6], [8], [9], [13], [15], [22], [28], [29], [30]). In [22], Kim defined the following q-Euler numbers…”

“…As q tends to 1 − in Theorem 10, we have the following Corollary. (for details, see [6], [8], [9], [13], [15], [22], [28], [29], [30]).…”