Let q be regarded as either a complex number q ∈ C or a p-adic number q ∈ Cp. If q ∈ C, then we always assume |q| < 1. If q ∈ Cp, we normally assume |1− q|p < p − 1 p−1 , which implies that q = exp(x log q) for |x|p ≤ 1. Here, | · |p is the p-adic absolute value in Cp with |p|p = 1 p . The q-basic natural number are defined by [n]q = 1−q 1−q = 1 + q + · · · + q , ( n ∈ N), and q-factorial are a...

This paper is an expanded version of my lecture delivered at the 97 Seattle summer research conference on finite fields. It gives a quick exposition of Dwork’s conjecture about p-adic meromorphic continuation of his unit root L-function arising from a family of algebraic varieties defined over a finite field of characteristic p. As a simple illustration, we discuss the classical example of the ...

In this paper, a set of transformations is defined on to. Some basic and naïve mathematical structure of is introduced. The concept of discrete dynamical systems through IVT and some further research scope of IVTs are highlighted. and 4]. A class of discrete transformations on named as Integral Value Transformations (IVT), corresponding to each of those CA rules, is introduced in the formation ...

In this article, I have studied the ultra discrete limit, which is currently studied in soliton theory, from point of view of valuation theory. A quantity obtained after taking the ultra discrete limit should be regarded as non-archimedean valuation, which is related to the p-adic valuation in number theory. The ultra discrete difference-difference equations, whose domain and range are given by...

To Dong-Sik Kim on his 62nd birthday We try to generalize an explicit construction for an orthonormal system of eigen-functions of the Vladimirov operator on L 2 (Q n p) which was originally given by Vladimirov (1988). Also the multiwavelet analysis can be considered as the p-adic spectral analysis in L 2 (Q n p), using in part the recent work of Kozyrev (2002). denote the ring of integers, the...

The integral representation algebra A(RG) is C ®z a(RG). When does a(RG) contain nontrivial nilpotent elements? Let | G\ = pn, where p\n, p prime. Denote by Zp the £-adic valuation ring in Q, and by Zp* its completion. Reiner has shown (i) If a = l , then A(ZPG) and A(Z*G) have no nonzero nilpotent elements (see [ l ] ) . (ii) If ce^2, and G has an element of order p, then both A(ZPG) and A{Z*G...

The Genocchi numbers and polynomials possess many interesting properties and are arising in many areas of mathematics and physics. Recently, many mathematicians have studied in the area of the q-Genocchi numbers and polynomials see 1–13 . In this paper, we construct a new type of q-Genocchi numbers G α,β n,q and polynomials G α,β n,q x with weight α and weak weight β. Throughout this paper, we ...

Using computer, a realistic study for new analogs of weighted q-Euler numbers and polynomials is very interesting. It is the aim of this paper to observe an interesting phenomenon of ‘scattering’ of the zeros of the weighted q-Euler polynomials E (α) n,q (x). The outline of this paper is as follows. In Section 2, we study weighted q-Euler polynomials E (α) n,q (x). In Section 3, we describe the...

In a remarkable 1993 paper, R. Sczech defined a group cocycle on GLn(Q) valued in a certain Q-vector space. This cocycle is constructed using conditionally convergent sums over lattices in Euclidean space and is called the Eisenstein cocycle. Sczech showed that certain specializations of the Eisenstein cocycle yield special values of the L-series of totally real fields. We will describe how one...