Generalized cyclotomic numbers with respect to prime powers

Abstract

In this paper, we study the generalized cyclotomic numbers with respect to the powers of odd primes, where at most one prime is of the form [Formula: see text]. We calculate the generalized cyclotomic numbers of order [Formula: see text] with respect to [Formula: see text] when the odd primes [Formula: see text] and the positive integers [Formula: see text] satisfy [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text].