Favard Constants

Author

Eric W. Weisstein
August 2, 2017

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/FavardConstants.nb.

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

©2017 Wolfram Research, Inc. except for portions noted otherwise

Sum

((4 )/π)(∑_(k=0)^∞)(((((((-1))^k)/(2 k+1))))^(r+1))==(2^(1-r)) Φ((((-1))^(r+1)),r+1,(1/2))

V6

(4/π)Sum[((((((-1)^k)/(2k+1))))^(r+1)),{k,0,∞}]

V7

(4/π)Sum[((((((-1)^k)/(2k+1))))^(r+1)),{k,0,∞}]

V11.2

(4/π)Sum[((((((-1)^k)/(2k+1))))^(r+1)),{k,0,∞}]

(((2^(1-r)) LerchPhi[(((-1))^(1+r)),1+r,(1/2)])/π)/.r->0

FullSimplify[(((2^(1-r)) LerchPhi[(((-1))^(1+r)),1+r,(1/2)])/π),Mod[r,2]==#]&/@{0,1}

FunctionExpand[(((2^(1-r)) LerchPhi[(((-1))^(1+r)),1+r,(1/2)])/π),Mod[r,2]==#]&/@{0,1}

FullSimplify[4/(((2π))^n)LerchPhi[(((-1))^n),n,(1/2)]-(4/(π^n))DirichletLambda[n],Mod[n,2]==0&&n>0]

Table[(π^(-n)) (-4 Zeta[n]+(2^(2-n)) (LerchPhi[1,n,(1/2)]+Zeta[n])),{n,2,10,2}]

FullSimplify[4/(((2π))^n)LerchPhi[(((-1))^n),n,(1/2)]-(4/(π^n))DirichletBeta[n],Mod[n,2]==1]

Values

FullSimplify[(((2^(1-r)) LerchPhi[#,1+r,(1/2)])/π)]&/@{-1,1}

Table[(2^(1-r)) LerchPhi[(((-1))^(1+r)),1+r,(1/2)]/π,{r,0,5}]

Table[4/(((2π))^n)LerchPhi[(((-1))^n),n,(1/2)],{n,21}]

Table[(4/(π^n))If[Mod[n,2]==0,DirichletLambda,DirichletBeta][n],{n,21}]