Solving Skolem's problem for the k–generalized Fibonacci sequence with negative indices

Abstract

The k–generalized Fibonacci sequence F(k) is the linearly recurrence sequence of order k whose first k terms are 0,…,0,1 and each term afterward is the sum of the preceding k terms. In this paper, we extend F(k) to negative indices obtaining a sequence denoted by H(k). Then, we solve Skolem's problem by finding all the zeros of H(k). To this end, we find very useful identities concerning H(k) similar to those obtained by others for F(k).