A matrix generalization of a theorem of Fine

TL;DR

This paper extends Fine's 1947 theorem by providing a matrix product formulation that counts binomial coefficients based on their p-adic valuation, and further generalizes to multinomial coefficients, with implications for p-regular sequences.

Contribution

It introduces a matrix product generalization of Fine's formula, enabling detailed counting of binomial coefficients by p-adic valuation and extending to multinomial coefficients.

Findings

Matrix product generalizes Fine's formula

Polynomial generating functions are p-regular

Extension to multinomial coefficients

Abstract

In 1947 Nathan Fine gave a beautiful product for the number of binomial coefficients $\binom{n}{m}$, for $m$ in the range $0 \leq m \leq n$, that are not divisible by $p$. We give a matrix product that generalizes Fine's formula, simultaneously counting binomial coefficients with $p$-adic valuation $\alpha$ for each $\alpha \geq 0$. For each $n$ this information is naturally encoded in a polynomial generating function, and the sequence of these polynomials is $p$-regular in the sense of Allouche and Shallit. We also give a further generalization to multinomial coefficients.