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.
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.
@article{JTNB_2009__21_1_145_0,
author = {Elsner, Carsten and Shimomura, Shun and Shiokawa, Iekata},
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},
mrnumber = {2537709},
zbl = {1233.11018},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.663/}
}
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 SP - 145 EP - 157 VL - 21 IS - 1 PB - Université Bordeaux 1 UR - http://www.numdam.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 Elsner, Carsten %A Shimomura, Shun %A Shiokawa, Iekata %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 http://www.numdam.org/articles/10.5802/jtnb.663/ %R 10.5802/jtnb.663 %G en %F JTNB_2009__21_1_145_0
Elsner, Carsten; Shimomura, Shun; Shiokawa, Iekata. Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 145-157. doi : 10.5802/jtnb.663. http://www.numdam.org/articles/10.5802/jtnb.663/
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