Products of extended binomial coefficients and their partial factorizations

TL;DR

This work defines extended factorials and extended binomial coefficients via Bhargava-type invariants and analyzes the products ${\overline{\overline{G}}}_n = \prod_{k=0}^n {n \choose k}_{\mathbb{Z},\mathbb{N}}$. The core approach expresses $\log {\overline{\overline{G}}}(n,x)$ as the difference of two radix-statistic sums, $\overline{A}(n,x)$ and $\overline{B}(n,x)$, and derives precise two-parameter asymptotics: $\log {\overline{\overline{G}}}_n = \frac{1}{2}n^2\log n + (\frac{1}{2}\gamma - \frac{3}{4})n^2 + O(n^{3/2}\log n)$ for the full product and $\log {\overline{\overline{G}}}(n,\alpha n) = f_{\overline{\overline{G}}}(\alpha)n^2\log n + g_{\overline{\overline{G}}}(\alpha)n^2 + O(n^{3/2}\log n)$ for partial factorizations, with explicit formulas for $f$ and $g$ and breakpoints at $\alpha=1/j$. The analysis hinges on detailed digit-sum statistics $d_b(n)$, $S_b(n)$, and $\overline{\nu}(n,b)$ across bases $2\le b\le x$, producing continuous limit scaling functions $f_{\overline{\overline{G}}}$ and $g_{\overline{\overline{G}}}$, and revealing how averaging over all bases smooths fluctuations (no RH needed). The results connect extended-binomial products to classic scaling phenomena previously observed for ordinary binomial products, while highlighting novel secondary terms and the role of Euler and Stieltjes constants. The work thus provides a comprehensive framework for understanding the large-scale behavior of these multiplicative combinatorial objects and their partial factorizations, with potential implications for related number-theoretic statistics and radix-based analyses.

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 $2\le b\le x$.

Figures (6)

• Figure 1.1: Graph of the limit scaling function $f_{{\overline{\overline{G}}}}(\alpha)$, $0 \le \alpha \le 1.$ Here $f_{{\overline{\overline{G}}}}(0)=0$ and $f_{{\overline{\overline{G}}}}(1)= \frac{1}{2}.$ The dashed line is $\tilde{f}(\alpha)= \frac{1}{2} \alpha$.
• Figure 1.2: Graph of the limit scaling function $g_{{\overline{\overline{G}}}}(\alpha)$, $0 \le \alpha \le 1$. Here $g_{{\overline{\overline{G}}}}(0)=0$ and $g_{{\overline{\overline{G}}}}(1) = \frac{1}{2} \gamma - \frac{3}{4} \approx - 0.46139$. The dashed curve is $\tilde{g}(\alpha)$.
• Figure 3.1: Graph of the limit scaling function $f_{{{\overline{B}}}}(\alpha)$, $0 \le \alpha \le 1$. Here $f_{{{\overline{B}}}}(0)=0$ and $f_{{{\overline{B}}}}(1)=1-\gamma\approx0.42278$.
• Figure 3.2: Graph of the limit scaling function $g_{{{\overline{B}}}}(\alpha)$, $0 \le \alpha \le 1$. Here $g_{{{\overline{B}}}}(0)=0$ and $g_{{{\overline{B}}}}(1)=\gamma+\gamma_1-1\approx -0.49560$.
• Figure 3.3: Graph of the limit scaling function $f_{{{\overline{A}}}}(\alpha)$, $0 \le \alpha \le 1$. Here $f_{{{\overline{A}}}}(0)=0$ and $f_{{{\overline{A}}}}(1)=\frac{3}{2}-\gamma\approx0.92278$.

Theorems & Definitions (78)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Definition 4.1
• Theorem 4.2