Numerical Computation of \prod_{n=1}^\infty (1 - tx^n)

TL;DR

The work develops a quadratically convergent method to compute $R(t,x)=\prod_{n\ge1}(1-tx^n)$ for complex $t$ and $x$ with $|x|<1$, using Euler’s expansion $R(t,x)=\sum_{m=0}^\infty \frac{(-t)^m x^{m(m+1)/2}}{\prod_{k=1}^m (1-x^k)}$. It establishes a two-sided bound on $R(1,x)$ that yields a sharp, $\approx9.1\%$-accurate bound on the Dedekind eta function $\eta(i y)$ for all $y>0$, and provides comprehensive a priori and a posteriori truncation analyses for the Euler-series computation. The paper also discusses asymptotic behavior of $R(t,x)$ as $x\to1$ for general $t$, and situates the method among other algorithms, highlighting the improved convergence near the unit circle. These results enable precise evaluation of $R(t,x)$ in combinatorics, modular forms, and statistical mechanics applications where $|x|$ approaches 1.

Abstract

I present and analyze a quadratically convergent algorithm for computing the infinite product \prod_{n=1}^\infty (1 - tx^n) for arbitrary complex t and x satisfying |x| < 1, based on the identity \prod_{n=1}^\infty (1 - tx^n) = \sum_{m=0}^\infty {(-t)^m x^{m(m+1)/2} \over (1-x)(1-x^2) ... (1-x^m)} due to Euler. The efficiency of the algorithm deteriorates as |x| \uparrow 1, but much more slowly than in previous algorithms. The key lemma is a two-sided bound on the Dedekind eta function at pure imaginary argument, η(iy), that is sharp at the two endpoints y=0,\infty and is accurate to within 9.1% over the entire interval 0 < y < \infty.

Figures (2)

• Figure 1: Graphs of $f(z)$ versus $z$ and $\log z$.
• Figure 2: Graphs of $g(z)$ versus $z$ and $\log z$.

Theorems & Definitions (13)

• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Corollary 2.4
• Proposition 2.5
• Proposition 2.6
• Corollary 2.7
• Lemma 3.1
• Lemma 3.2
• Proposition 3.3