infinite series with Hypergeometric functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T07:00:11Z http://mathoverflow.net/feeds/question/83555 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83555/infinite-series-with-hypergeometric-functions infinite series with Hypergeometric functions Remy 2011-12-15T19:34:01Z 2011-12-16T23:16:59Z <p>Can we get a closed form for the series</p> <p>$\sum^\infty_{k=0} \frac{ t^k}{k!} \Gamma(k+a)\Gamma(k+\frac{1}{2}){}_2F_1(k+a,k+\frac{1}{2};n+1,x)$</p> <p>any hints or clues are welcomed.</p> http://mathoverflow.net/questions/83555/infinite-series-with-hypergeometric-functions/83669#83669 Answer by Tony Cahill for infinite series with Hypergeometric functions Tony Cahill 2011-12-16T23:05:52Z 2011-12-16T23:16:59Z <p>I too wonder about convergence. You can rewrite it as <code>$$\Gamma \left( a\right) \Gamma \left( 1/2\right) \sum_{k=0}^{\infty }\sum_{j=0}^{\infty } \frac{\left( a\right) _{j+k} \left( 1/2\right) _{j+k}}{\left( n+1\right) _{j}}\frac{t^{k}}{k!}\frac{x^{j}}{j!};$$</code> if you had an additional Pochhammer term indexed by k in the denominator, it would be Appel's $F_{4}$ function.</p>