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Asymptotic expansions for ratios of products of gamma functions. (English) Zbl 1020.33001
The following asymptotic formula for a ratio of products of gamma functions is proved: \[ \begin{split} \frac{\Gamma(a_1+n)\Gamma(a_2+n)\cdots\Gamma(a_{p+1}+n)}{\Gamma(b_1+n)\cdots\Gamma(b_p+n)\Gamma(-s_p+n)}\\ =1+\sum_{m=1}^M\frac{(a_1+s_p)_m(a_2+s_p)_m}{(1)_m(1+s_p-n)_m} \sum_{k=0}^m\frac{(-m)_k}{(a_1+s_p)_k(a_2+s_p)_k}A_k^{(p)}+O(n^{-M-1})\text{ as }n\to\infty,\end{split} \] where \(s_p=b_1+\cdots+b_p-a_1-\cdots-a_{p+1}\) and \(A_k^{(p)}\) are determined as follows: Let the generalized hypergeometric function \({}_{p+1}F_p(a_1,\ldots,a_{p+1};b_1,\ldots,b_p;z)\) be analytically continued near \(z=1\) and be represented as \[ \begin{split} \frac{\Gamma(a_1)\Gamma(a_2)\cdots\Gamma(a_{p+1})}{\Gamma(b_1)\cdots\Gamma(b_p)}_{p+1}F_p(a_1,\ldots,a_{p+1};b_1,\ldots,b_p;z)\\ =\sum_{m=0}^\infty g_m(0)(1-z)^m+(1-z)^{s_p}\sum_{m=0}^\infty g_m(s_p)(1-z)^m.\end{split} \] Then \[ g_m(s_p)=(-1)^m\frac{(a_1+s_p)_m(a_2+s_p)_m\Gamma(-s_p-m)}{(1)_m}\sum_{k=0}^m\frac{(-m)_k}{(a_1+s_p)_k(a_2+s_p)_k}A_k^{(p)}. \]

MSC:
33B15 Gamma, beta and polygamma functions
33C20 Generalized hypergeometric series, \({}_pF_q\)
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