Golden Binomials and Carlitz Characteristic Polynomials

Abstract

The golden binomials, introduced in the golden quantum calculus, have expansion determined by Fibonomial coefficients and the set of simple zeros given by powers of Golden ratio. We show that these golden binomials are equivalent to Carlitz characteristic polynomials of certain matrices of binomial coefficients. It is shown that trace invariants for powers of these matrices are determined by Fibonacci divisors, quantum calculus of which was developed very recently.