Bailey's Theorem

Author

Eric W. Weisstein
August 20, 2008

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

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

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

(((((Γ(m+(1/2)))/(Γ(m)))))^2) (∑_(k=0)^(n-1))(((((((2 k-1)!!)/((2 k)!!))))^2)/(k+m))=(((((Γ(n+(1/2)))/(Γ(n)))))^2) (∑_(k=0)^(m-1))(((((((2 k-1)!!)/((2 k)!!))))^2)/(k+n))

Apply[Subtract,Table[{(Gamma[(m+1/2)]/Gamma[m])^2Sum[((2k-1)!!/(2k)!!)^2/(m+k),{k,0,n-1}],
(Gamma[(n+1/2)]/Gamma[n])^2Sum[((2k-1)!!/(2k)!!)^2/(n+k),{k,0,m-1}]
},{m,5},{n,5}],{2}]

e1=FullSimplify[(Gamma[(m+1/2)]/Gamma[m])^2Sum[((2k-1)!!/(2k)!!)^2/(m+k),{k,0,n-1}],(m|n)∈Integers]

e2=FullSimplify[(Gamma[(n+1/2)]/Gamma[n])^2Sum[((2k-1)!!/(2k)!!)^2/(n+k),{k,0,m-1}],(m|n)∈Integers]

FullSimplify[e1-e2]