On Fibonacci Polynomial Expressions for Sums of $m$th Powers, their implications for Faulhaber's Formula and some Theorems of Fermat

Abstract

Denote by $Σn^m$ the sum of the $m$-th powers of the first $n$ positive integers $1^m+2^m+\ldots +n^m$. Similarly let $Σ^r n^m$ be the $r$-fold sum of the $m$-th powers of the first $n$ positive integers, defined such that $Σ^0 n^{m}=n^m$, and then recursively by $Σ^{r+1} n^{m}=Σ^{r} 1^{m}+Σ^{r} 2^{m}+\ldots + Σ^{r} n^{m}$. During the early 17th-century, polynomial expressions for the sums $Σ^r n^m$ and their factorisation and polynomial basis representation properties were examined by Johann Faulhaber, who published some remarkable theorems relating to these $r$-fold sums in his Academia Algebrae (1631). In this paper we consider families of polynomials related to the Fibonacci and Lucas polynomials which naturally lend themselves to representing sums and differences of integer powers, as well as $Σ^r n^m$. Using summations over polynomial basis representations for monomials we generalise this sum for any polynomial $q(x)\in\mathbb{Q}[x]$, encountering some interesting coefficient families in the process. Other results include using our polynomial expressions to state some theorems of Fermat, where in particular, we obtain an explicit expression for the quotient in Fermat's Little Theorem. In the final two sections we examine these sums in the context of the three pairs of Stirling number varieties and binomial decompositions. We also derive an expression for $Σn^m$ in terms of the Stirling numbers and Pochhammer symbols, leading to a seemingly new relation between the Stirling numbers and the Bernoulli numbers.