Gauss's Digamma Theorem

Author

Eric W. Weisstein
March 26, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Sums

2 (∑_k^(⌈q/2⌉-1))ln(sin(((π k)/q))) cos(((2 π k p)/q))-ln(2 q)-(1/2) π cot(((π p)/q))-ℽ==(PolyGamma2[0, (p/q)])

-EulerGamma-Log[2q]-(1/2)π Cot[(p/q)π]+2Sum[Cos[((2π p k)/q)]Log[Sin[((π k)/q)]],{k,Ceiling[q/2]-1}]

-EulerGamma-Log[2q]-(1/2)π Cot[(p/q)π]+2Sum[Cos[((2π p k)/q)]Log[Sin[((π k)/q)]],{k,Ceiling[q/2]-1},Assumptions->0<p<q&&(p|q)∈Integers]

<<Utilities`Simplify`

Function[{p,q},Print[{p/q,v=-EulerGamma-(1/2) π Cot[((p π)/q)]-Log[2 q]+2Sum[Cos[((2π p k)/q)]Log[Sin[((π k)/q)]],{k,Ceiling[q/2]-1}]//LogExpand//Simplify//LogExpand,{v,PolyGamma[0,p/q]}//N}]]@@@{{1,2},{1,3},{2,3},{1,4},{3,4},{1,6},{5,6},{1,1}};

Check

pq=Select[Flatten[Table[{p,q},{q,5},{p,q-1}],1],CoprimeQ@@#&]

Subtract@@@Function[{p,q},{
((Derivative[1][Gamma][p/q])/(Gamma[p/q])),
-EulerGamma-Log[2q]-(1/2)π Cot[(p/q)π]+2Sum[Cos[((2π p k)/q)]Log[Sin[((π k)/q)]],{k,Ceiling[q/2]-1}]
}]@@@pq//FullSimplify//Union