Variations on an error sum function for the convergents of some powers of e

2014/08/10 by Jean‐Paul Allouche, Thomas Baruchel, Allouche, Jean-Paul +1 · 1 citation
Computer Science · Mathematics · #11A55 #11B75 #11B83 #11J70 #Coding theory and cryptography #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Numerical Methods and Algorithms

paper · pdf · doi:10.48550/arxiv.1408.2206

openalex publication_date 2014/08/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Several years ago the second author playing with different "recognizers of real constants", e.g., the LLL algorithm, the Plouffe inverter, etc. found empirically the following formula. Let p_n/q_n denote the nth convergent of the continued fraction of the constant e, then ∑_{n ≥ 0} |q_n e - p_n| = (e)/(4) (- 1 + 10 ∑_{n ≥ 0} ((-1)ⁿ)/((n+1)! (2n² + 7n + 3))). The purpose of the present paper is to prove this formula and to give similar formulas for some powers of e.

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• Weighted error-sum identities for periodic continued fractions and their generalizations

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