hypergeometric transformation I came across the following ${}_3F_2$ hypergeometric polynomial:
$$
{}_3F_2\left(\left.\begin{array}{c} 1,1,-n\\ 2, -1-2n  \end{array}\right| -x\right)
$$
for some large $x > 0$. I am wondering if there is some transformation or identity that can change the above function to some hypergeometric function with small $x$ (though it may turn a polynomial into an infinite series)?
 A: We can specialize:
$ _{u+1}F_{v}\left[\begin{array}{cc}
-n & \alpha_{1},\ldots,\alpha_{u}\\
 & \beta_{1},\ldots,\beta_{v}
\end{array};z\right] = \frac{\left(\alpha_{1}\right)_{n}\cdots\left(\alpha_{u}\right)_{n}}{\left(\beta_{1}\right)_{n}\cdots\left(\beta_{v}\right)_{n}}\left(-z\right)^{n}\cdot_{v+1}F_{u}\left[\begin{array}{cc}
-n & 1-\beta_{1}-n,\ldots,1-b_{v}-n\\
 & 1-\alpha_{1}-n,\ldots,1-\alpha_{v}-n
\end{array};\frac{\left(-1\right)^{u+v}}{z}\right]$
To
$_{3}F_{2}\left[\begin{array}{c}
-n,1,1\\
2,-1-2n
\end{array};-z\right]=\frac{\Gamma(n+1)\cdot\Gamma\left(n+1\right)}{\Gamma\left(n+2\right)\cdot\Gamma\left(-1-2n\right)}\left(z\right)^{n}\cdot_{3}F_{2}\left[\begin{array}{cc}
-n & ,-1-n,n+2\\
 & -n,-n
\end{array};-\frac{1}{z}\right]
 $
Which should make your life easier for large Z.
This is Eq 1.8 in: 
Some General Families of Generating Functions for the Laguerre Polynomials 
http://www.sciencedirect.com/science/article/pii/S0022247X83711376
and of course 

Lucy's standard work at Eq. 2.2.3.2
Generalized HyperGeometric Functions
Although my edition seems to have an error in the equation. Reading the text ahead of it gives the above formula.

And 16.2.3 in the DLMF
http://dlmf.nist.gov/16.2.E3
