Existence of Pisano-like periods for Fibonacci-type polynomial sequences

Abstract

We consider the existence of Pisano-like periods for Fibonacci-type polynomial sequences. First we show that the existence of a Pisano-like period is characterized by the divisibility of some Fibonacci-type polynomial by the modulus. We also show for a subclass of Fibonacci-type polynomials that such moduli are in one-to-one correspondence with the Fibotomic polynomials. Lastly, we show that the same divisibility criterion is sufficient for Lucas-type polynomials, but is not necessary.