Volume 8, 2002, Number 1

We detail and study the periods of the inverses of integers in order to understand the behavior of periodic arithmetic sequences whose terms are the digits of the inverses of integers. More precisely, by introducing the rank of a prime number we show how to compute the period of the inverse of prime numbers of variable rank. Furthermore, we prove some properties and a number of statements concerning the residue class 10(mod n), (10, n) = 1, which lead to define the degree of a prime number. In addition, by using both the rank and the degree of a prime number we compute the periods of the inverses of powers of prime numbers. Finally, we study the behavior of the simple recurring decimals and we establish our main result which shows how to compute the period and its length of the inverse of an integer.

In the recent paper, we defined the h-extension of q-Bernoulli number by using multiple p-adic q-integral and constructed the h-extension of complex analytic q-L-series which interpolates the h-extension of q-Bernoulli numbers, cf. [2], [4], [5]. The purpose of this paper is to construct a h-extension of p-adic q-L-function which interpolates the h-extension of q-Bernoulli numbers at non-positive integers.