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 [Wolfram's page][1]:

$$(x)_n \sim \frac{\sqrt{2\pi}}{\Gamma(x)} e^{-n}n^{x+n-\frac12}(1+O(\frac1n)) \qquad n \to \infty$$


  [1]: http://functions.wolfram.com/GammaBetaErf/Pochhammer/introductions/FactorialBinomials/ShowAll.html