Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values

Carsten Elsner[1]; Shun Shimomura[2]; Iekata Shiokawa[2]

  • [1] Fachhochschule für die Wirtschaft University of Applied Sciences Freundallee 15 30173 Hannover, Germany
  • [2] Department of Mathematics Keio University 3-14-1 Hiyoshi, Kohoku-ku Yokohama 223-8522 Japan

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 1, page 145-157
  • ISSN: 1246-7405

Abstract

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We present asymptotic representations for certain reciprocal sums of Fibonacci numbers and of Lucas numbers as a parameter tends to a critical value. As limiting cases of our results, we obtain Euler’s formulas for values of zeta functions.

How to cite

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Elsner, Carsten, Shimomura, Shun, and Shiokawa, Iekata. "Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values." Journal de Théorie des Nombres de Bordeaux 21.1 (2009): 145-157. <http://eudml.org/doc/10867>.

@article{Elsner2009,
abstract = {We present asymptotic representations for certain reciprocal sums of Fibonacci numbers and of Lucas numbers as a parameter tends to a critical value. As limiting cases of our results, we obtain Euler’s formulas for values of zeta functions.},
affiliation = {Fachhochschule für die Wirtschaft University of Applied Sciences Freundallee 15 30173 Hannover, Germany; Department of Mathematics Keio University 3-14-1 Hiyoshi, Kohoku-ku Yokohama 223-8522 Japan; Department of Mathematics Keio University 3-14-1 Hiyoshi, Kohoku-ku Yokohama 223-8522 Japan},
author = {Elsner, Carsten, Shimomura, Shun, Shiokawa, Iekata},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Fibonacci number; Lucas number; asymptotic representation; zeta function},
language = {eng},
number = {1},
pages = {145-157},
publisher = {Université Bordeaux 1},
title = {Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values},
url = {http://eudml.org/doc/10867},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Elsner, Carsten
AU - Shimomura, Shun
AU - Shiokawa, Iekata
TI - Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 1
SP - 145
EP - 157
AB - We present asymptotic representations for certain reciprocal sums of Fibonacci numbers and of Lucas numbers as a parameter tends to a critical value. As limiting cases of our results, we obtain Euler’s formulas for values of zeta functions.
LA - eng
KW - Fibonacci number; Lucas number; asymptotic representation; zeta function
UR - http://eudml.org/doc/10867
ER -

References

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  3. D. Duverney, Ke. Nishioka, Ku. Nishioka, and I. Shiokawa,Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers. Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), 140–142. Zbl0902.11029MR1487578
  4. C. Elsner, S. Shimomura, and I. Shiokawa,Algebraic relations for reciprocal sums of Fibonacci numbers. Acta Arith. 130 (2007), 37–60. Zbl1132.11036MR2354148
  5. C. Elsner, S. Shimomura, and I. Shiokawa,Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers. Ramanujan J. 17 (2008), 429–446. Zbl1233.11081MR2456843
  6. C. Elsner, S. Shimomura, and I. Shiokawa,Algebraic relations for reciprocal sums of even terms in Fibonacci numbers. To appear in St. Petersburg Math. J. Zbl06406739MR2405627
  7. H. Hancock,Theory of Elliptic Functions. Dover, New York, 1958. Zbl0084.07302
  8. A. Hurwitz and R. Courant,Vorlesungen über allgemeine Funktionentheorie und ellipti-sche Funktionen. Springer, Berlin, 1925. 
  9. C. G. J. Jacobi,Fundamenta Nova Theoriae Functionum Ellipticarum. Königsberg, 1829. 
  10. Yu. V. Nesterenko,Modular functions and transcendence questions. Mat. Sb. 187 (1996), 65–96; English transl. Sb. Math. 187 (1996), 1319–1348. Zbl0898.11031MR1422383
  11. E. T. Whittaker and G. N. Watson,Modern Analysis, 4th ed. Cambridge Univ. Press, Cambridge, 1927. 
  12. I. J. Zucker,The summation of series of hyperbolic functions. SIAM J. Math. Anal. 10 (1979), 192–206. Zbl0411.33001MR516762

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