Click here to flash read.
arXiv:2404.00332v3 Announce Type: replace
Abstract: 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 linear recurrences, as well as explicit formulas for certain combinatorial sequences, including the Pell numbers and central binomial coefficients. For example, we present the formula: $\binom{2n}{n} = \left((4^n + 1)^{2n} \mod (4^{n(n+1)} + 1)\right) \mod (4^n-1)$. We also discover an unusual representation for the real $r$th roots of positive integers, characterized as the limit of a modular congruence. 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.
No creative common's license