Approximation of values of hypergeometric functions by restricted rationals

Elsner, Carsten, Komatsu, Takao, and Shiokawa, Iekata. "Approximation of values of hypergeometric functions by restricted rationals." Journal de Théorie des Nombres de Bordeaux 19.2 (2007): 393-404. <http://eudml.org/doc/249954>.

@article{Elsner2007,
abstract = {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\hspace\{4.44443pt\}(\@mod \; 2) $ solving the Pythagorean equation $x^2 + y^2 = z^2 $ satisfy\[|y\sinh (1/s) - x| \,\gg \frac\{\log \log y\}\{\log y\} \,\,.\]Conversely, for every $s=1,2,\dots $ there are infinitely many coprime integers $x,y $, such that\[|y\sinh (1/s) - x| \,\ll \frac\{\log \log y\}\{\log y\} \]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.},
affiliation = {FHDW Hannover, University of Applied Sciences Freundallee 15 D-30173 Hannover, Germany; Faculty of Science and Technology Hirosaki University Hirosaki, 036-8561, Japan; Department of Mathematics Keio University Hiyoshi 3-14-1 Yokohama, 223-8522, Japan},
author = {Elsner, Carsten, Komatsu, Takao, Shiokawa, Iekata},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {irrationality measure; hypergeometric function},
language = {eng},
number = {2},
pages = {393-404},
publisher = {Université Bordeaux 1},
title = {Approximation of values of hypergeometric functions by restricted rationals},
url = {http://eudml.org/doc/249954},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Elsner, Carsten
AU - Komatsu, Takao
AU - Shiokawa, Iekata
TI - Approximation of values of hypergeometric functions by restricted rationals
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 2
SP - 393
EP - 404
AB - 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\hspace{4.44443pt}(\@mod \; 2) $ solving the Pythagorean equation $x^2 + y^2 = z^2 $ satisfy\[|y\sinh (1/s) - x| \,\gg \frac{\log \log y}{\log y} \,\,.\]Conversely, for every $s=1,2,\dots $ there are infinitely many coprime integers $x,y $, such that\[|y\sinh (1/s) - x| \,\ll \frac{\log \log y}{\log y} \]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.
LA - eng
KW - irrationality measure; hypergeometric function
UR - http://eudml.org/doc/249954
ER -