Rising factorials

Question

Let $a^{\bar{n}}$ be a rising factorial or Pochammer symbol and ${\Gamma(a)}$ be the Gamma function. I want to ask if $0^{\bar{0}}$=$1$ or not. As you know there is a relation between the rising factorial and The Gamma Function: $a^{\bar{n}}=\frac{\Gamma(a+n)}{\Gamma(a)}$ . I want to ask when a=n=0 can we claim $0^{\bar{0}}$=$\frac{\Gamma(0)}{\Gamma(0)}$ =$1$? I looked at some books. But I am a little bit complicated. Thanks for your answers and references.

Answer 1

The rising factorial is defined as $a^{\overline{n}} = \prod_{i=1}^n (a+i-1)$, so in particular, $a^{\overline{0}} = \prod_{i=1}^0 (a+i-1)$, which is an empty product - but an empty product is always 1, so so $a^{\overline{0}} = 1$.