Hypergeometic function. Questions.

[LagrangeEuler]

Legendre polynomial is defined as
$P_n(x)=_2F_1(-n,n+1;1,\frac{1-x}{2})$
Pochammer symbols are defined as $(a)_n=\frac{\Gamma(a+n)}{\Gamma(a)}$. If I undestand well
P_n(x)=_2F_1(-n,n+1;1,\frac{1-x}{2}) =\sum^{\infty}_{k=0}\frac{(-n)_k(n+1)_k}{(1)_kk!}x^k
I am not sure what happens with
(-n)_k=\frac{\Gamma(-n+k)}{\Gamma(-n)}
because $\Gamma$ diverge for negative integers.

 

[Geofleur]

Even though $\Gamma$ diverges for negative integers, the ratio of gamma functions evaluated at different integer arguments does not necessarily diverge. First, note that $\Gamma(n) = (n-1)!$. Now write out the terms as follows:

$\frac{\Gamma(-n+k)}{\Gamma(-n)} = \frac{(k-n-1)!}{(-n-1)!} = \frac{(k-n-1)(k-n-2)\cdots (k-n-(k-1))(k-n-k)(k-n-(k+1))\cdots }{(-n-1)(-n-2)(-n-3)\cdots}.$

Therefore, we have

$\frac{\Gamma(-n+k)}{\Gamma(-n)} = \frac{(k-n-1)(k-n-2)\cdots (-n+1)(-n)(-n-1)\cdots }{(-n-1)(-n-2)(-n-3)\cdots} = (k-n-1)(k-n-2)\cdots(-n+1)(-n)$,

and this last result is finite.