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Ramanujan phi-Function


The two-argument Ramanujan function is defined by

phi(a,n)=1+2sum_(k=1)^(n)1/((ak)^3-ak)
(1)
=1-1/a(H_(-1/a)+H_(1/a)+2H_n-H_(n-1/a)-H_(n+1/a)).
(2)

The one-argument function phi(a) is then defined as the limiting sum of phi(a,n) as n->infty,

phi(a)=lim_(n->infty)phi(a,n)
(3)
=1+2sum_(k=1)^(infty)1/((ak)^3-ak)
(4)
=-1/a[psi_0(1/a)+psi_0(1-1/a)+2gamma]
(5)
=1-1/a(H_(-1/a)+H_(1/a)),
(6)

where psi_0(x) is the digamma function, gamma is the Euler-Mascheroni constant, and H_nu is a harmonic number. The values of phi(n) for n=2, 3, ... are

phi(2)=2ln2
(7)
phi(3)=ln3
(8)
phi(4)=3/2ln2
(9)
phi(5)=1/5sqrt(5)lnphi+1/2ln5
(10)
phi(6)=1/2ln3+2/3ln2,
(11)

where phi is the golden ratio.


See also

Harmonic Number, Ramanujan g- and G-Functions, Ramanujan Theta Functions, Tau Function

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Cite this as:

Weisstein, Eric W. "Ramanujan phi-Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RamanujanPhi-Function.html

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