Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 145-157.

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.

Nous présentons les représentations asymptotiques pour certaines sommes des réciproques des nombres de Fibonacci et des nombres de Lucas quand un paramètre tend vers une valeur critique. Comme cas limite de nos résultats, nous obtenons les formules d’Euler pour les valeurs des fonctions de zeta.

DOI: 10.5802/jtnb.663
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
@article{JTNB_2009__21_1_145_0,
     author = {Carsten Elsner and Shun Shimomura and Iekata Shiokawa},
     title = {Asymptotic representations for {Fibonacci} reciprocal sums and {Euler{\textquoteright}s} formulas for zeta values},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {145--157},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {1},
     year = {2009},
     doi = {10.5802/jtnb.663},
     zbl = {1233.11018},
     mrnumber = {2537709},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.663/}
}
TY  - JOUR
AU  - Carsten Elsner
AU  - Shun Shimomura
AU  - Iekata Shiokawa
TI  - Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2009
SP  - 145
EP  - 157
VL  - 21
IS  - 1
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.663/
DO  - 10.5802/jtnb.663
LA  - en
ID  - JTNB_2009__21_1_145_0
ER  - 
%0 Journal Article
%A Carsten Elsner
%A Shun Shimomura
%A Iekata Shiokawa
%T Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values
%J Journal de théorie des nombres de Bordeaux
%D 2009
%P 145-157
%V 21
%N 1
%I Université Bordeaux 1
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.663/
%R 10.5802/jtnb.663
%G en
%F JTNB_2009__21_1_145_0
Carsten Elsner; Shun Shimomura; Iekata Shiokawa. Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 145-157. doi : 10.5802/jtnb.663. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.663/

[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Dover, New York, 1965.

[2] D. Duverney, Ke. Nishioka, Ku. Nishioka, and I. Shiokawa, Transcendence of Jacobi’s theta series. Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 202–203. | MR | Zbl

[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. | MR | Zbl

[4] C. Elsner, S. Shimomura, and I. Shiokawa, Algebraic relations for reciprocal sums of Fibonacci numbers. Acta Arith. 130 (2007), 37–60. | MR | Zbl

[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. | MR

[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. | MR

[7] H. Hancock, Theory of Elliptic Functions. Dover, New York, 1958.

[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. | MR | Zbl

[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. | MR | Zbl

Cited by Sources: