# Ramanujan change of base

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##### 1: 20.11 Generalizations and Analogs
###### §20.11(iii) Ramanujan’s Change of Base
These results are called Ramanujan’s changes of base. …
##### 2: 8.11 Asymptotic Approximations and Expansions
If $x=a+(2a)^{\frac{1}{2}}y$ and $a\to+\infty$, then
8.11.10 $P\left(a+1,x\right)=\tfrac{1}{2}\operatorname{erfc}\left(-y\right)-\frac{1}{3}% \sqrt{\frac{2}{\pi a}}(1+y^{2})e^{-y^{2}}+O\left(a^{-1}\right),$
8.11.11 $\gamma^{*}\left(1-a,-x\right)=x^{a-1}\left(-\cos\left(\pi a\right)+\frac{\sin% \left(\pi a\right)}{\pi}\left(2\sqrt{\pi}F\left(y\right)+\frac{2}{3}\sqrt{% \frac{2\pi}{a}}\left(1-y^{2}\right)\right)e^{y^{2}}+O\left(a^{-1}\right)\right),$
##### 3: Bibliography B
• R. J. Baxter (1981) Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.
• A. Berkovich and B. M. McCoy (1998) Rogers-Ramanujan Identities: A Century of Progress from Mathematics to Physics. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 163–172.
• B. C. Berndt, S. Bhargava, and F. G. Garvan (1995) Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.
• B. C. Berndt (1989) Ramanujan’s Notebooks. Part II. Springer-Verlag, New York.
• B. C. Berndt (1991) Ramanujan’s Notebooks. Part III. Springer-Verlag, Berlin-New York.
• ##### 4: 15.8 Transformations of Variable
With $\zeta=e^{\ifrac{2\pi\mathrm{i}}{3}}(1-z)/\left(z-e^{\ifrac{4\pi\mathrm{i}}{3}}\right)$
15.8.32 $\frac{\left(1-z^{3}\right)^{a}}{\left(-z\right)^{3a}}\left(\frac{1}{\Gamma% \left(a+\frac{2}{3}\right)\Gamma\left(\frac{2}{3}\right)}F\left({a,a+\frac{1}{% 3}\atop\frac{2}{3}};z^{-3}\right)+\frac{e^{\frac{1}{3}\pi\mathrm{i}}}{z\Gamma% \left(a\right)\Gamma\left(\frac{4}{3}\right)}F\left({a+\frac{1}{3},a+\frac{2}{% 3}\atop\frac{4}{3}};z^{-3}\right)\right)=\frac{3^{\frac{3}{2}a+\frac{1}{2}}e^{% \frac{1}{2}a\pi\mathrm{i}}\Gamma\left(a+\frac{1}{3}\right)(1-\zeta)^{a}}{2\pi% \Gamma\left(2a+\frac{2}{3}\right)(-\zeta)^{2a}}F\left({a+\frac{1}{3},3a\atop 2% a+\frac{2}{3}};\zeta^{-1}\right),$ $|z|>1$, $|\operatorname{ph}\left(-z\right)|<\frac{1}{3}\pi$.