×

Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers. (English) Zbl 1233.11081

Summary: In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers \(\sum^{\infty}_{n=1} F^{-1}_{2n-1} , \sum^{\infty}_{n=1} F^{-2}_{2n-1} , \sum^{\infty}_{n=1} F^{-3}_{2n-1}\) and write each \(\sum^{\infty}_{n=1} F^{-s}_{2n-1}\) \((s\geq 4)\) as an explicit rational function of these three numbers over \(\mathbb Q\). Similar results are obtained for various series including the reciprocal sums of odd terms in Lucas numbers.

MSC:

11J81 Transcendence (general theory)
11J91 Transcendence theory of other special functions
33E05 Elliptic functions and integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Duverney, D., Nishioka, Ke., Nishioka, Ku., Shiokawa, I.: Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers. Proc. Jpn. Acad. Ser. A Math. Sci. 73, 140–142 (1997) · Zbl 0902.11029 · doi:10.3792/pjaa.73.140
[2] Elsner, C., Shimomura, S., Shiokawa, I.: Algebraic relations for reciprocal sums of Fibonacci numbers. Preprint (2006) · Zbl 1155.11041
[3] Nesterenko, Y.V.: Modular functions and transcendence questions. Mat. Sb. 187, 65–96 (1996). English translation Sb. Math. 187, 1319–1348 (1996) · Zbl 0898.11031
[4] Zucker, I.J.: The summation of series of hyperbolic functions. SIAM J. Math. Anal. 10, 192–206 (1979) · Zbl 0411.33001 · doi:10.1137/0510019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.