On the Fermionic p-adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials
نویسندگان
چکیده
Throughout this paper, let p be an odd prime number. The symbol, p, p , and p denote the ring of p-adic integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of p , respectively. Let be the set of natural numbers and ∪ {0}. Let νp be the normalized exponential valuation of p with |p|p p−νp p 1/p. Note that p {x | |x|p ≤ 1} lim← N /p p. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ , or p-adic number q ∈ p . If q ∈ , we normally assume |q| < 1, and if q ∈ p , we always assume |1 − q|p < 1. We say that f is uniformly differentiable function at a point a ∈ p and write f ∈ UD p , if the difference quotient Ff x, y f x − f y / x − y has a limit f ′ a as x, y → a, a . For f ∈ UD p , the fermionic p-adic q-integral on p is defined as
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