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Euler's Hypergeometric Transformations

{}_2F_1(a,b;c;z) = \int_0^1 {t^{b-1}(1-t)^{c-b-1}\over (1-tz)^a}\,dt,
\end{displaymath} (1)

where ${}_2F_1(a,b;c;z)$ is a Hypergeometric Function. The solution can be written using the Euler's transformations
$\displaystyle t$ $\textstyle \to$ $\displaystyle t$ (2)
$\displaystyle t$ $\textstyle \to$ $\displaystyle 1-t$ (3)
$\displaystyle t$ $\textstyle \to$ $\displaystyle (1-z-tz)^{-1}$ (4)
$\displaystyle t$ $\textstyle \to$ $\displaystyle {1-t\over 1-tz}$ (5)

in the equivalent forms
$ {}_2F_1(a,b;c;z) = (1-z)^{-a} \,{}_2F_1(a,c-b;c;z/(z-1))\quad$ (6)
$ = (1-z)^{-b} \,{}_2F_1(c-a,b;c;z/(z-1))\quad$ (7)
$ = (1-z)^{c-a-b} \,{}_2F_1(c-a,c-b;c;z).\quad$ (8)

See also Hypergeometric Function


Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 585-591, 1953.

© 1996-9 Eric W. Weisstein