## Mark Daniel Ward - Asymptotic Rational Approximation To Pi: Solution of an "Unsolved Problem'' Posed By Herbert Wilf

dmtcs:2777 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10) - https://doi.org/10.46298/dmtcs.2777
Asymptotic Rational Approximation To Pi: Solution of an "Unsolved Problem'' Posed By Herbert Wilf

Authors: Mark Daniel Ward 1

• 1 Department of Statistics

The webpage of Herbert Wilf describes eight Unsolved Problems. Here, we completely resolve the third of these eight problems. The task seems innocent: find the first term of the asymptotic behavior of the coefficients of an ordinary generating function, whose coefficients naturally yield rational approximations to $\pi$. Upon closer examination, however, the analysis is fraught with difficulties. For instance, the function is the composition of three functions, but the innermost function has a non-zero constant term, so many standard techniques for analyzing function compositions will completely fail. Additionally, the signs of the coefficients are neither all positive, nor alternating in a regular manner. The generating function involves both a square root and an arctangent. The complex-valued square root and arctangent functions each rely on complex logarithms, which are multivalued and fundamentally depend on branch cuts. These multiple values and branch cuts make the function extremely tedious to visualize using Maple. We provide a complete asymptotic analysis of the coefficients of Wilf's generating function. The asymptotic expansion is naturally additive (not multiplicative); each term of the expansion contains oscillations, which we precisely characterize. The proofs rely on complex analysis, in particular, singularity analysis (which, in turn, rely on a Hankel contour and transfer theorems).

Volume: DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10)
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: analytic combinatorics,asymptotic analysis,rational approximation,generating function,singularity analysis,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]
Funding:
Source : OpenAIRE Graph
• Emerging Frontiers of Science of Information; Funder: National Science Foundation; Code: 0939370