Polynomial quotient rings and Kronecker substitution for deriving combinatorial identities

Abstract

We introduce a new approach for generating combinatorial identities and formulas by the application of Kronecker substitution to polynomial expansions within quotient rings. Our main result enables the derivation of elementary arithmetic formulas for many C-recursive integer sequences directly from their characteristic polynomials. As sample applications, we present new formulas for the Pell numbers and central binomial coefficients, which are famous integer sequences. These applications lead us to the discovery of a new and unusual formula for the real $n$-th roots of positive integers, $\sqrt[n]{a}$, characterized as the limit of a quotient involving modular exponentiations. From this limit formula we conjecture a fixed-length elementary closed form expression for $\lfloor \sqrt[n]{a} \rfloor$.