Hurwitzian Continued Fractions Containing a Repeated Constant and An Arithmetic Progression

DOI10.1137/130926092zbMATH Open1379.11006arXiv1211.2494OpenAlexW2091281179MaRDI QIDQ3192173

Gabor Hetyei

Publication date: 26 September 2014

Published in: SIAM Journal on Discrete Mathematics (Search for Journal in Brave)

Abstract: We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves combinatorics and formal Laurent series. Using very little analysis we can express their limits in terms of (modified) Bessel functions and Fibonacci polynomials. The limit formula is a generalization of Lehmer's theorem that implies the continuous fraction expansions of $e$ and $an(1)$, and it can also be derived from Lehmer's work using Fibonacci polynomial identities. We completely characterize those implementations of our limit formula for which the parameter of each Bessel function is the half of an odd integer, allowing them to be replaced with elementary functions.

Full work available at URL: https://arxiv.org/abs/1211.2494

zbMATH Keywords

Bessel functionsFibonacci polynomialsLucas polynomialsconvergentsHurwitzian continued fraction

Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) Continued fractions (11A55) Continued fractions; complex-analytic aspects (30B70)

Related Items (1)

Hurwitz continued fractions with confluent hypergeometric functions

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