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

## TL;DR

This paper introduces new Jacobi-type continued fraction expansions for binomial coefficients and establishes novel congruences modulo any integer, expanding the understanding of binomial coefficient properties beyond prime moduli.

## Contribution

It presents new Jacobi-type J-fraction expansions for binomial coefficients and derives general congruences modulo any integer, including composite numbers, with exact formulas from convergents.

## Key findings

- New Jacobi-type J-fraction expansions for binomial coefficients
- Generalized congruences for binomial coefficients modulo any integer
- Exact formulas from convergents of the continued fractions

## Abstract

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 $n$.