Determinants containing powers of polynomial sequences

TL;DR

This paper derives new determinant identities for matrices with entries as powers of polynomial sequences satisfying recurrence relations, extending known identities for Fibonacci, Lucas, and orthogonal polynomials.

Contribution

It introduces generalized determinant identities for polynomial sequences, broadening the scope of previous specific cases involving Fibonacci and Lucas polynomials.

Findings

Derived identities for determinants of polynomial power matrices

Unified framework for Fibonacci, Lucas, and orthogonal polynomials

Extended previous determinant identities to broader polynomial classes

Abstract

We derive identities for the determinants of matrices whose entries are (rising) powers of (products of) polynomials that satisfy a recurrence relation. In particular, these results cover the cases for Fibonacci polynomials, Lucas polynomials and certain orthogonal polynomials. These identities naturally generalize the determinant identities obtained by Alfred, Carlitz, Prodinger, Tangboonduangjit and Thanatipanonda.