Approximation of values of hypergeometric functions by restricted rationals
Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 2, pp. 393-404.

We compute upper and lower bounds for the approximation of hyperbolic functions at points 1/s (s=1,2,) by rationals x/y, such that x,y satisfy a quadratic equation. For instance, all positive integers x,y with y0(mod2) solving the Pythagorean equation x 2 +y 2 =z 2 satisfy

|ysinh(1/s)-x|loglogylogy.

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

|ysinh(1/s)-x|loglogylogy

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,) par des rationnels x/y, tels que x,y satisfassent une équation quadratique. Par exemple, tous les entiers positifs x,y avec y0(mod2), solutions de l’équation de Pythagore x 2 +y 2 =z 2 , satisfont

|ysinh(1/s)-x|loglogylogy.

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

|ysinh(1/s)-x|loglogylogy

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.

Received:
Published online:
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
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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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.593/

[1] C. Elsner, On arithmetic properties of the convergents of Euler’s number. Colloq. Math. 79 (1999), 133–145. | Zbl: 0930.11048

[2] C. Elsner, On rational approximations by Pythagorean numbers. Fibonacci Quart. 41 (2003), 98–104. | MR: 1990517 | Zbl: 1028.11042

[3] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers. Fifth edition, Clarendon Press, Oxford, 1979. | MR: 568909 | Zbl: 0020.29201

[4] A. Khintchine, Kettenbrüche. B.G.Teubner Verlagsgesellschaft, 1956. | MR: 80630

[5] T. Komatsu, Arithmetical properties of the leaping convergents of e 1/s . Tokyo J. Math. 27 (2004), 1–12. | MR: 2060069 | Zbl: 1075.11004

[6] L. J. Mordell, Diophantine equations. Academic Press, London and New York, 1969. | MR: 249355 | Zbl: 0188.34503

[7] O. Perron, Die Lehre von den Kettenbrüchen. Chelsea, New York, 1950. | MR: 37384 | Zbl: 0041.18206

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