The Power series is $ $ R_n(x)=\Sigma_{p=2}^{n+1}\frac{(p-1)(2n-p)!}{n!(n+1-p)!}x^p$ $

It says the following:

For large $ n$ , the asymptotic form of $ R_n(1/\beta)$ can be obtained by looking at the values of $ p$ which dominate the sum

- $ p$ of order $ 1$ for $ \beta > 1/2$ ,
- $ p$ of order $ \sqrt n$ for $ \beta = 1/2$ , and
- $ p \propto (1 – 2\beta)/(1-\beta)$ for $ \beta < i$

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