How to explain hypergeometric $2F_1[1+m,n,2+m,-2]$?

Question

Question as title showed. What expression it represents? Many thanks.

Answer 1

Hypergeometric2F1 is the Gauss hypergeometric function: http://mathworld.wolfram.com/HypergeometricFunction.html

$_2F_1(1+m,0;2+m;-2)=1$

In case of $n=$negative integer, it reduces to polynomial fractions :

$_2F_1(1+m,-1;2+m;-2)=\frac{3m+4}{m+2}$

$_2F_1(1+m,-2;2+m;-2)=\frac{9m^2+33m+26}{(m+2)(m+3)}$

$_2F_1(1+m,-3;2+m;-2)=\frac{27m^3+189m^2+396m+240}{(m+2)(m+3)(m+4)}$

In case of $n=$positive integer, the simplest closed form is the hyperfeometric function itself.

On integral form : $$_2F_1(1+m,n;2+m;-2)=\frac{\Gamma(m+2)}{\Gamma(m+2+n)\Gamma(n)}\int_0^1 (1-t)^{1+m-n} t^{n-1} (1+2t)^{-m-1}dt$$