  We introduce a framework for generating combinatorial identities by applying
Kronecker substitution to polynomial expansions within quotient rings. We apply
this methodology to derive a general formula for C-recursive integer sequences,
as well as explicit formulas for certain combinatorial sequences, including the
Pell numbers and central binomial coefficients. We also discover an unusual
representation for the real $n$-th roots of positive integers $\sqrt[n]{a}$,
characterized as the limit of a modular expression. The theorems and results
presented provide a theoretical basis for future research on the
interconnections between Kronecker substitution, polynomial ring expansions,
and their applications in combinatorial sequences and beyond.
