Approximation of values of hypergeometric functions by restricted rationals

Carsten Elsner; Takao Komatsu; Iekata Shiokawa

doi:10.5802/jtnb.593

Carsten Elsner ¹; Takao Komatsu ²; Iekata Shiokawa 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, Volume 19 (2007) no. 2, pp. 393-404

Abstract (Original language)
Résumé

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.

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.

MR Zbl

DOI: 10.5802/jtnb.593

Author's affiliations:

Carsten Elsner ¹; Takao Komatsu ²; Iekata Shiokawa 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

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     title = {Approximation of values of hypergeometric functions by restricted rationals},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {393--404},
     year = {2007},
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Carsten Elsner; Takao Komatsu; Iekata Shiokawa. Approximation of values of hypergeometric functions by restricted rationals. Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 2, pp. 393-404. doi: 10.5802/jtnb.593

References
Cited by

[1] C. Elsner, On arithmetic properties of the convergents of Euler’s number. Colloq. Math. 79 (1999), 133–145. | Zbl

[2] C. Elsner, On rational approximations by Pythagorean numbers. Fibonacci Quart. 41 (2003), 98–104. | Zbl | MR

[3] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers. Fifth edition, Clarendon Press, Oxford, 1979. | Zbl | MR

[4] A. Khintchine, Kettenbrüche. B.G.Teubner Verlagsgesellschaft, 1956. | MR

[5] T. Komatsu, Arithmetical properties of the leaping convergents of $e^{1 / s}$ . Tokyo J. Math. 27 (2004), 1–12. | Zbl | MR

[6] L. J. Mordell, Diophantine equations. Academic Press, London and New York, 1969. | Zbl | MR

[7] O. Perron, Die Lehre von den Kettenbrüchen. Chelsea, New York, 1950. | Zbl | MR

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