Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers $h\ge 2$

Abstract

We prove two new forms of Jacobi-type J-fraction expansions generating the binomial coefficients, $\left(\genfrac{}{}{0ex}{}{x+n}{n}\right)$ and $\left(\genfrac{}{}{0ex}{}{x}{n}\right)$, over all $n\ge 0$. Within the article we establish new forms of integer congruences for these binomial coefficient variations modulo any (prime or composite) $h\ge 2$ and compare our results with existing known congruences for the binomial coefficients modulo primes $p$ and prime powers $p^k$. We also prove new exact formulas for these binomial coefficient cases from the expansions of the $h^{th}$ convergent functions to the infinite J-fraction series generating these coefficients for all $n$.