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

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.

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.

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DOI : https://doi.org/10.5802/jtnb.593
@article{JTNB_2007__19_2_393_0,
     author = {Carsten Elsner and Takao Komatsu and Iekata Shiokawa},
     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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.593/}
}
Carsten Elsner; Takao Komatsu; Iekata Shiokawa. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.593/

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