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arXiv · 1702.01374

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

Maxie D. Schmidt
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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 $n$.

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Maxie D. Schmidt. 2017-02-05. Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers $h \geq 2$. https://arxiv.org/abs/1702.01374

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