Euler's Hypergeometric Transformations

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

where ${}_2F_1(a,b;c;z)$ is a Hypergeometric Function. The solution can be written using the Euler's transformations

$$t \to t$$ (2)

$$t \to 1-t$$ (3)

$$t \to (1-z-tz)^{-1}$$ (4)

$$t \to {1-t\over 1-tz}$$ (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

References

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