On the period of the continued fraction for values of the square root of power sums

Abstract

The present paper proves that if for a power sum $\alpha$ over $\ZZ$ the length of the period of the continued fraction for $\sqrt{\alpha(n)}$ is constant for infinitely many even (resp. odd) $n$, then $\sqrt{\alpha(n)}$ admits a functional continued fraction expansion for all even (resp. odd) $n$, except finitely many; in particular, for such $n$, the partial quotients can be expressed by power sums of the same kind.