Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 31/Miscellaneous Properties

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Miscellaneous Properties

$31.14$: Gauss's Hypergeometric Theorem

$\map F {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$


$31.15$: Derivative of Gaussian Hypergeometric Function

$\map {\dfrac \d {\d x} } {\map F {a, b; c; x} } = \dfrac {a b} c \map F {a + 1, b + 1; c + 1; x} $


$31.16$: Euler's Integral Representation of Hypergeometric Function

$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{- a} \rd t$


$31.17$: Euler's Transformation

$\ds \map F {a, b; c; x} = \paren {1 - x}^{c - a - b} \map F {c - a, c - b; c; x}$