Some divisibility properties of binomial and q-binomial coefficients

Abstract

We first prove that if $a$ has a prime factor not dividing $b$ then there are infinitely many positive integers $n$ such that $\left(\genfrac{}{}{0ex}{}{an+bn}{an}\right)$ is not divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover, we provide some new divisibility properties of binomial coefficients: for example, we prove that $\left(\genfrac{}{}{0ex}{}{12n}{3n}\right)$ and $\left(\genfrac{}{}{0ex}{}{12n}{4n}\right)$ are divisible by $6n-1$, and that $\left(\genfrac{}{}{0ex}{}{330n}{88n}\right)$ is divisible by $66n-1$, for all positive integers $n$. As we show, the latter results are in fact consequences of divisibility and positivity results for quotients of $q$-binomial coefficients by $q$-integers, generalizing the positivity of $q$-Catalan numbers. We also put forward several related conjectures.

16 pages, add a note that Conjectures 7.2 and 7.3 are proved, to appear in J. Number Theory

References in corpus (3)

• Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
• A q-rious positivity
• Proof of Sun's conjecture on the divisibility of certain binomial sums

Cited by in corpus (5)

• Proof of Sun's conjectures on integer-valued polynomials
• Stable maps to Looijenga pairs
• Factors of sums involving $q$-binomial coefficients and powers of $q$-integers
• On Parity Unimodality of $q$-Catalan Polynomials
• Proof of a conjecture related to divisibility properties of binomial coefficients