Approximation of values of hypergeometric functions by restricted rationals

Elsner, Carsten; Komatsu, Takao; Shiokawa, Iekata

doi:10.5802/jtnb.593

Elsner, Carsten ¹ ; Komatsu, Takao ² ; Shiokawa, Iekata ³

¹ FHDW Hannover, University of Applied Sciences Freundallee 15 D-30173 Hannover, Germany
² Faculty of Science and Technology Hirosaki University Hirosaki, 036-8561, Japan
³ Department of Mathematics Keio University Hiyoshi 3-14-1 Yokohama, 223-8522, Japan

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 393-404.

Résumé
Abstract

Nous calculons des bornes supérieures et inférieures pour l’approximation de fonctions hyperboliques aux points $1 / s$ $(s = 1, 2, \dots)$ par des rationnels $x / y$ , tels que $x, y$ satisfassent une équation quadratique. Par exemple, tous les entiers positifs $x, y$ avec $y \equiv 0 (mod 2)$ , solutions de l’équation de Pythagore $x^{2} + y^{2} = z^{2}$ , satisfont

| y sinh (1 / s) - x | ≫ \frac{log log y}{log y} .

Réciproquement, pour chaque $s = 1, 2, \dots$ , il existe une infinité d’entiers $x, y$ , premiers entre eux, tels que

| y sinh (1 / s) - x | ≪ \frac{log log y}{log y}

et $x^{2} + y^{2} = z^{2}$ soient réalisés simultanément avec $z$ entier. Une généralisation à l’approximation de $h (e^{1 / s})$ , pour $h (t)$ fonction rationnelle, est incluse.

We compute upper and lower bounds for the approximation of hyperbolic functions at points $1 / s$ $(s = 1, 2, \dots)$ by rationals $x / y$ , such that $x, y$ satisfy a quadratic equation. For instance, all positive integers $x, y$ with $y \equiv 0 (mod 2)$ solving the Pythagorean equation $x^{2} + y^{2} = z^{2}$ satisfy

| y sinh (1 / s) - x | ≫ \frac{log log y}{log y} .

Conversely, for every $s = 1, 2, \dots$ there are infinitely many coprime integers $x, y$ , such that

| y sinh (1 / s) - x | ≪ \frac{log log y}{log y}

and $x^{2} + y^{2} = z^{2}$ hold simultaneously for some integer $z$ . A generalization to the approximation of $h (e^{1 / s})$ for rational functions $h (t)$ is included.

MR Zbl

DOI : 10.5802/jtnb.593

Affiliations des auteurs :

Elsner, Carsten ¹ ; Komatsu, Takao ² ; Shiokawa, Iekata ³

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     author = {Elsner, Carsten and Komatsu, Takao and Shiokawa, Iekata},
     title = {Approximation of values of hypergeometric functions by restricted rationals},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {393--404},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {2},
     year = {2007},
     doi = {10.5802/jtnb.593},
     zbl = {1167.11026},
     mrnumber = {2394893},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.593/}
}

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Elsner, Carsten; Komatsu, Takao; Shiokawa, Iekata. Approximation of values of hypergeometric functions by restricted rationals. Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 393-404. doi : 10.5802/jtnb.593. http://www.numdam.org/articles/10.5802/jtnb.593/

Bibliographie
Cité par

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