We prove two new forms of Jacobi-type J-fraction expansions generating the binomial coefficients, $\binom{ x+n}{ n}$ and $\binom{ x}{ n}$, over all $n \geq 0$. Within the article we establish new forms of integer congruences for these binomial coefficient variations modulo any (prime or composite) $h \geq 2$ 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 $n$.
Categories: math.CO