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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)?

user58955
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    The {1},{2} combination can be interpreted (and eliminated) as integration and the Hypergeometric transformed to 2F1 * C ; C linear, more or less, in n,x. Then you can have exact results. – rrogers May 30 '16 at 21:19

1 Answers1

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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

rrogers
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