Numerical Computation of ∏_n=1^∞ (1 - txⁿ)

2002/12/03 by Alan D. Sokal, Sokal, Alan D.
Computer Science · Mathematics · #11F20 #11P82 #33D99 #33F05 (Primary) 05A30 #65D20 #82B23 (Secondary) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical and Theoretical Analysis #Numerical Analysis (math.NA) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.math/0212035

openalex publication_date 2002/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

I present and analyze a quadratically convergent algorithm for computing the infinite product ∏_n=1^∞ (1 - txⁿ) for arbitrary complex t and x satisfying |x| < 1, based on the identity ∏_n=1^∞ (1 - txⁿ) = ∑_m=0^∞ (-t)^m x^m(m+1)/2 \over (1-x)(1-x²) ... (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,∞ and is accurate to within 9.1% over the entire interval 0 < y < ∞.

Related

• Representations of analytic functions as infinite products and their application to numerical computations
• An Asymptotic Form of the Generating Function ∏_k=1^∞ (1+x^k/k)
• On the Modular Behaviour of the Infinite Product (1-x)(1-xq)(1-xq²)(1-xq³)...
• Fast multi-precision computation of some Euler products
• Infinite products with algebraic numbers