Thomae's Theorem

Author

Eric W. Weisstein
May 25, 2003

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

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

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

Identity

thomae=With[{s=e+f-a-b-c},
{((Gamma[a])/(Gamma[e]Gamma[f]))HypergeometricPFQ[{a,b,c},{e,f},1],
((Gamma[s])/(Gamma[s+b]Gamma[s+c]))HypergeometricPFQ[{s,e-a,f-a},{s+b,s+c},1]
}]

Check

Exact

FullSimplify[Subtract@@thomae,Re[a]>0&&Re[e+f-a-b-c]>0]

Numerical

Length[vals=Select[Table[Random[Complex,{-2-2I,2+2I},20],{100},{5}],Re[#[[1]]]>0&&Re[#[[4]]+#[[5]]-(#[[1]]+#[[2]]+#[[3]])]>0&]]

Random

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v4.2

Subtract@@thomae/.Thread[{a,b,c,e,f}->#]&/@vals

v5.0

Subtract@@thomae/.Thread[{a,b,c,e,f}->#]&/@vals

Sondow's Special Case

May 21, 2003

id={
HypergeometricPFQ[{1,a,b},{a+1,d},1]/a,HypergeometricPFQ[{1,d-b,d-a},{d-b+1,d},1]/(d-b)}

Table[
Subtract@@id/.Thread[{a,b,c,d}->Table[Random[Real,{0,2}],{4}]],
{5}]