Slater's Formula

Author

Eric W. Weisstein
May 19, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Formula

(Subscript[, 4])(Subscript[F, 3])(a,b,-n,(a/2)+1;(a/2),a-b+1,a+n+1;1)==(((Subscript[(a+1), n]) (Subscript[(((a+1)/2)-b), n]))/((Subscript[(((a+1)/2)), n]) (Subscript[(a-b+1), n])))

HypergeometricPFQ[{a,b,-n,a/2+1},{a/2,1+a-b,1+a+n},1]==(Pochhammer[1+a,n]*Pochhammer[(1+a)/2-b,n])/(Pochhammer[(1+a)/2,n]*Pochhammer[1+a-b,n])

Generalization

(Subscript[, 4])(Subscript[F, 3])(a,b,c,(a/2)+1;(a/2),a-b+1,a-c+1;1)==((Γ(((a+1)/2)) Γ(a-b+1) Γ(a-c+1) Γ(((a+1)/2)-b-c))/(Γ(a+1) Γ(((a+1)/2)-b) Γ(((a+1)/2)-c) Γ(a-b-c+1)))/;Re(a-2 b-2 c)>-1

HypergeometricPFQ[{a,b,c,a/2+1},{a/2,1+a-b,1+a-c},1]==(Gamma[(1+a)/2]*Gamma[1+a-b]*Gamma[1+a-c]*Gamma[(1+a)/2-b-c])/(Gamma[1+a]*Gamma[(1+a)/2-b]*Gamma[(1+a)/2-c]*Gamma[1+a-b-c])/;Re[a-2*b-2*c]>-1