Convert a polynomial to Hypergeometric equation

Question

How can I convert this polynomial to Hypergeometric equation.

$$Pn=n!\sum_{r=0}^{\frac{n}{2} } (-1)^r\frac{(p-1)!}{(p-n+r-1)!} \frac{1}{r! (n-2r)!}2x^{n-2r}$$

Answer 1

I think that you need to consider two different cases and almost then use the definition of the gaussian hypergeometric function to get $$P_{2n}=\frac{2 (p-1)! }{(p-2 n-1)!}\,x^{2 n} \, _2F_1\left(-n+\frac{1}{2},-n;p-2 n;-\frac{4}{x^2}\right)$$ $$P_{2n+1}=\frac{2 (p-1)! }{(p-2 n-2)!}\,x^{2 n+1} \, _2F_1\left(-n-\frac{1}{2},-n;p-2 n-1;-\frac{4}{x^2}\right)$$