Gauss's Hypergeometric Theorem

Author

Eric W. Weisstein
September 27, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/GausssHypergeometricTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GausssHypergeometricTheorem.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Theorem

FullSimplify[Hypergeometric2F1[a,b,c,1]-Pochhammer[c-b,-a]/Pochhammer[c,-a],Re[c-a-b]>0]

FunctionExpand[%]

FullSimplify[Hypergeometric2F1[a,b,c,1]-((Gamma[c] Gamma[-a-b+c])/(Gamma[-a+c] Gamma[-b+c])),Re[c-a-b]>0]

Numeric Check

Hypergeometric2F1[a,b,c,1]-((Gamma[c] Gamma[-a-b+c])/(Gamma[-a+c] Gamma[-b+c]))/.Thread[{a,b,c}->#]&/@Function[f,Select[f,Re[#[[3]]-#[[1]]-#[[2]]]>0&]][Table[Random[Complex],{200},{3}]]//Chop