Products of extended binomial coefficients and their partial factorizations

TL;DR

This paper investigates the properties and asymptotic behavior of a new class of extended binomial coefficient products, generalizing Pascal's triangle, using advanced factorial theory and radix expansion statistics.

Contribution

It introduces extended binomial coefficients based on generalized factorials and derives asymptotic formulas for their products, extending Bhargava's factorial theory.

Findings

Logarithm of the product approximated by quadratic functions in n

Asymptotic formulas include a power-saving remainder term

Analysis involves radix expansion statistics of integers

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, $n!_S=\prod_p\nu_n(S,p)$ 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 $n!_{\mathbb{Z},\mathbb{N}}=\prod_{b\ge2}b^{\alpha_n(\mathbb{Z},b)}$ including all $b\ge2$, with associated extended binomial coefficients $\binom{n}{k}_{\mathbb{Z},\mathbb{N}}$, yielding $\overline{\overline{G}}_n$. We have $\overline{\overline{G}}_n=\prod_{b=2}^nb^{\overline{\nu}(n,b)}$ and the partial factorizations $\overline{\overline{G}}(n,x)=\prod_{b=2}^{\lfloor x\rfloor}b^{\overline{\nu}(n,b)}$. This paper shows $\log\overline{\overline{G}}(n,\alpha n)$ is well approximated by $f_{\overline{\overline{G}}}(\alpha)n^2\log n+g_{\overline{\overline{G}}}(\alpha)n^2$ as $n\to\infty$ for limit functions $f_{\overline{\overline{G}}}(\alpha)$ and $g_{\overline{\overline{G}}}(\alpha)$ defined for all $0\le\alpha\le1$. 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 $2\le b\le x$.