Morley's Formula

Author

Eric W. Weisstein
August 5, 2002

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

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

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

Comments

Something is wrong with convergence here.  I need to go back and figure out what...

General Case

m1=Sum[(Pochhammer[m,k]/k!)^3,{k,0,∞}]

Limit[m1,m->1]

Table[m1/.m->n,{n,5}]

m2=Gamma[1-3/2m]/Gamma[1-m/2]^3Cos[m π/2]

Limit[m2,m->3]

Table[m2/.m->n,{n,5}]

Power 1

FullSimplify[Sum[(Pochhammer[m,k]/k!),{k,0,Infinity}]]

Power 2

FullSimplify[Sum[(Pochhammer[m,k]/k!)^2,{k,0,Infinity}]]

FullSimplify[HypergeometricPFQ[{m,m},{1},1]]

FullSimplify[{Hypergeometric2F1[m,m,1,1],(((4^(-m)) Gamma[(1/2)-m])/((Sqrt[π]) Gamma[1-m]))}]

Table[%,{m,5}]

Power 3

Sum[Pochhammer[m,k]^3/k!^3,{k,0,5}]

FullSimplify[Sum[(Pochhammer[m,k]/k!)^3,{k,0,Infinity}]]

FullSimplify[-((3 Gamma[1+(m/2)] Gamma[-((3 m)/2)] Sin[m π])/(2 π ((Gamma[1-(m/2)])^2)))-Gamma[1-3/2m]/Gamma[1-m/2]^3 Cos[Pi m/2]]

FullSimplify[HypergeometricPFQ[{m,m,m},{1,1},1]]

Power 4

FullSimplify[Sum[(Pochhammer[m,k]/k!)^4,{k,0,Infinity}]]

FullSimplify[HypergeometricPFQ[{m,m,m,m},{1,1,1},1]]

Power 5

FullSimplify[Sum[(Pochhammer[m,k]/k!)^5,{k,0,Infinity}]]