most recent 30 from http://mathoverflow.net 2013-06-19T21:39:36Z http://mathoverflow.net/feeds/question/109229 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109229/hypergeometric-function-2f-1-n-r12 Campello 2012-10-09T14:12:25Z 2012-10-11T21:28:13Z

<p>The hypergeometric function $_2F_1(-n;-r;1;2)$ appears in many different situations. For instance, it counts the number of integer points within a sphere in the $l_1$ norm, i.e., </p> <p>$$_2F_1(-n;-r;1;2) = \mbox{#} \lbrace x \in \mathbb{Z}^n : |x_1| + \cdots + |x_n| \leq r \rbrace $$</p> <p>and another formula for $_2F_1(-n;-r;1;2)$ is given by</p> <p>$$_2F_1(-n;-r;1;2) = \sum_{i=1}^{\min\lbrace{n,r}\rbrace}{n \choose i}{r \choose i}2^i$$</p> <p>Does anyone know an asymptotic formula for this function when $n$ is large?</p> <p>I know there are some closed formulas for the Gaussian hypergeometric (i.e. $_2F_1(a,b;2,1)$) and in some other cases, but haven't find any clue in this case.</p>