# Integrating over simplices, and the Gamma function

(Polya, Szego, Problems and Theorems in Analysis). Pick a sequence of positive numbers, $\lambda_1,\ldots,\lambda_p$, and set $f_k(z)= \sum_{\nu \geqslant 0} \nu^{\lambda_k -1} z^\nu$. Now write $f(z)= f_1(z)\cdots f_p(z) = \sum_{\nu \geqslant 1} a_\nu z^\nu$. Show that

$\lim\limits_{\nu \to\infty}\dfrac{a_\nu}{\nu^{\lambda_1+\cdots+\lambda_p-1}}=\int_{\Delta^{p-1}} F(x)dx$

where $F(x) = x_1^{\lambda_1-1}\cdots x_{p-1}^{\lambda_{p-1}-1} (1-x_1-\cdots-x_{p-1})^{\lambda_p-1}$ and $\Delta^{p-1}$  is the canonical $p-1$-dimensional simplex. In fact, the limit is $\dfrac{\Gamma(\lambda_1)\cdots \Gamma(\lambda_p)}{\Gamma(\lambda_1+\cdots+\lambda_p)}$.