Golden Binomials and Carlitz Characteristic Polynomials

TL;DR

This paper explores the relationship between golden binomials, Fibonacci numbers, and Carlitz characteristic polynomials, revealing new connections in quantum calculus and matrix invariants.

Contribution

It establishes the equivalence between golden binomials and Carlitz characteristic polynomials, linking Fibonacci divisors with matrix trace invariants in quantum calculus.

Findings

Golden binomials are equivalent to Carlitz characteristic polynomials.

Trace invariants relate to Fibonacci divisors.

Quantum calculus of Fibonacci divisors is connected to matrix properties.

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.