PEMODIC ASPECTS OF SEQUENCES GENERATED BY TWO SPECIAL MAPPINGS

Let β = $\beta = \frac{q} {p}$ be a fixed rational number, where p and q are positive integers with 2 ≤ p < q and gcd(p, q) = 1. Consider two real-valued functions σ(x) = βx mod 1 and τ(x) = βx mod 1. For each positive integer n, let $s\left( n \right) = \sigma \left( n \right) = \frac{{s{{\left( n \right)}_1}}} {p} + \cdots + \frac{{s{{\left( n \right)}_n}}} {{{p^n}}}andt\left( n \right) = {\tau ^n}\left( 1 \right) = \frac{{t{{\left( n \right)}_1}}} {p} + \cdots + \frac{{t{{\left( n \right)}_n}}} {{{p^n}}}$ be the p-ary representation. In this paper, we study the periods of both sequences ${S_k} = \left\{ {s{{\left( {n + k} \right)}_n}} \right\}\begin{array}{*{20}{c}} \infty \\ {n = 1} \\ \end{array} and{T_k} = \left\{ {t{{\left( {n + k} \right)}_n}} \right\}\begin{array}{*{20}{c}} \infty \\ {n = 1} \\ \end{array} $ for any non-negative integer k.