Periodicities of a class of infinite integer sequences modulo M
Let \(\{u_ n\}\) be an infinite sequence of integers with generating function A(x). If \(A(x)=1/f(x)\), where f(x) is a polynomial with integer coefficients and \(f(0)=1\), the \(\{u_ n\}\) is a linear recursive sequence and it is periodic modulo any integer \(M>1\). In this paper the author develops a treatment of investigation of periodicity of these sequences via their generating functions. From general theorems many old and new results are obtained. Let p be a prime and let \(f(x)=c_ kx^ k+...+c_ 1x+c_ 0\) with conditions \(c_ 0=1\) and \(\gcd (c_ k,p)=1\). Suppose f(x) is irreducible modulo p. Among others it is proved: (i) If \(\alpha\) is a zero of f(x) modulo \(p^ N\), then the minimum period of the sequence generated by 1/f(x) modulo \(p^ N\) is the order of \(\alpha\) modulo \(p^ N\). (ii) Let \(\lambda\) be the minimum period of the sequence generated by 1/f(x) modulo p. Then for \(p^{b-1}<t\leq p^ b,\) where \(b\geq 1\), the minimum period of the sequence generated by \(1/(f(x))^ t\) modulo \(p^ N\) is \(\lambda p^{N+b-1}.\)
scientific article; zbMATH DE number 4075164
scientific article