Products of extended binomial coefficients and their partial factorizations

Abstract

This paper studies properties of the integer sequence $\overline{\overline{G}}_n=\prod_{k=0}^n\binom{n}{k}_{\mathbb{Z},\mathbb{N}}$ which is analogous to $\overline{G}_n=\prod_{k=0}^n\binom{n}{k}$, the product of the elements of the $n$-th row of Pascal's triangle. Here $\binom{n}{k}_{\mathbb{Z},\mathbb{N}}$ is an extended binomial coefficient, defined in the paper, constructed using an extended version of M. Bhargava's theory of generalized factorials. In 1996 M. Bhargava introduced a generalization of the factorial function, $n!_S=\prod_pν_n(S,p)$ in terms of their prime factorization, and defines associated binomial coefficients. The last two authors extended Bhargava's invariants further to define such invariants attached to each integer $b\ge2$. One obtains extended factorials and extended binomial coefficients, and the maximal extension defines extended factorials $n!_{\mathbb{Z},\mathbb{N}}=\prod_{b\ge2}b^{α_n(\mathbb{Z},b)}$ including all $b\ge2$, with associated extended binomial coefficients $\binom{n}{k}_{\mathbb{Z},\mathbb{N}}$, yielding $\overline{\overline{G}}_n$. We have $\overline{\overline{G}}_n=\prod_{b=2}^nb^{\overlineν(n,b)}$ and the partial factorizations $\overline{\overline{G}}(n,x)=\prod_{b=2}^{\lfloor x\rfloor}b^{\overlineν(n,b)}$. This paper shows $\log\overline{\overline{G}}(n,αn)$ is well approximated by $f_{\overline{\overline{G}}}(α)n^2\log n+g_{\overline{\overline{G}}}(α)n^2$ as $n\to\infty$ for limit functions $f_{\overline{\overline{G}}}(α)$ and $g_{\overline{\overline{G}}}(α)$ defined for all $0\leα\le1$. The remainder term has a power saving in $n$. The main results are deduced from study of functions $\overline{A}(n,x)$ and $\overline{B}(n,x)$ that encode statistics of the base $b$ radix expansions of the integer $n$ (and smaller integers), where the base $b$ ranges over all integers $2\le b\le x$.