Large finite products of small fractions

TL;DR

This paper investigates the asymptotic behavior of large finite products of small fractions involving a function similar to sine, establishing a relation between the product and a power of n with a constant factor.

Contribution

It provides a new asymptotic formula for products of functions resembling sine near zero, extending understanding of their growth as n increases.

Findings

The product behaves asymptotically like a constant times n to the power of (a-b)/c.

The result holds when m grows linearly with n.

The analysis applies to functions similar to sine near zero.

Abstract

Fix positive reals $a,b,c,d$, and let $h(x)$ be a real function behaving sort of like $\sin x$ near 0. Then, provided $m$ grows linearly with $n$. there exists a positive constant $C$ such that$$ \prod_{j=0}^m\frac{h\left((cj+a)\frac{d}{n}\right)}{h\left((cj+b)\frac{d}{n}\right)}\sim C n^{\frac{a-b}c}. $$