Ramanujan Phi-Function

Author

Eric W. Weisstein
December 10, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/RamanujanPhi-Function.nb.

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

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

Definition

RamanujanPhi[a_]:=1-(1/a)(HarmonicNumber[-1/a]+HarmonicNumber[1/a])

Two-Arg

1+2Sum[(1/((a k)^3-a k)),{k,n}]

FullSimplify[%]

TraditionalForm[%]

One-Arg

1+2Sum[(1/((a k)^3-a k)),{k,∞}]

FullSimplify[%]

TraditionalForm[%]

<<Utilities`Simplify`

ϕ(2)

RamanujanPhi[2]//FullSimplify//LogExpand

ϕ(4)

RamanujanPhi[3]//FullSimplify

ϕ(5)

RamanujanPhi[4]//FullSimplify//LogExpand

ϕ(5)

RamanujanPhi[5]//FullSimplify//LogExpand

FunctionExpand[%]//FullSimplify

FullSimplify[Subtract@@{%,(1/5)(Sqrt[5])Log[GoldenRatio]+(1/2)Log[5]}]

ϕ(6)

RamanujanPhi[6]//FullSimplify//LogExpand//Expand