What is the Stirling formula for x(x+1)(x+2)...(x+n-1)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:50:26Z http://mathoverflow.net/feeds/question/58231 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58231/what-is-the-stirling-formula-for-xx1x2-xn-1 What is the Stirling formula for x(x+1)(x+2)...(x+n-1)? 7-adic 2011-03-12T04:54:35Z 2011-03-12T05:48:38Z <p>Let x be a complex number.</p> <p>What is the Stirling formula for x(x+1)(x+2)...(x+n-1) when n goes to infinity?</p> http://mathoverflow.net/questions/58231/what-is-the-stirling-formula-for-xx1x2-xn-1/58235#58235 Answer by S. Carnahan for What is the Stirling formula for x(x+1)(x+2)...(x+n-1)? S. Carnahan 2011-03-12T05:48:38Z 2011-03-12T05:48:38Z <p>It looks like you want a formula for the asymptotics of the Pochhammer symbol $(x)_n$ as $n \to \infty$. One such formula is provided about halfway down <a href="http://functions.wolfram.com/GammaBetaErf/Pochhammer/introductions/FactorialBinomials/ShowAll.html" rel="nofollow">Wolfram's page</a>:</p> <p>$$(x)_n \sim \frac{2\pi}{\Gamma(x)} e^{-n}n^{x+n-\frac12}(1+O(\frac1n)) \qquad n \to \infty$$</p>