Derivative of Dirichlet Beta function
We prove the following infinite product which turned out to be a modified version of the Dirichlet Beta function derivative. 13√35√57√79√911√1113√1315√15…=(πeγΓ(34)4)π4
Proof
Take logarithm in both sides −ln(3)3+ln(5)5−ln(7)7+ln(9)9−ln(11)11+ln(13)13−...=π4[ln(π)−γ−4lnΓ(34)]
Then, we want to prove
∞∑n=1(−1)n+1ln(2n−1)(2n−1)=π4[ln(π)−γ−4lnΓ(34)]
Consider the Dirichlet beta function
β(v)=∞∑n=1(−1)n+1(2n−1)v
Then
β′(v)=∞∑n=1(−1)nln(2n−1)(2n−1)v
Therefore we want to find
−β′(1)=∞∑n=1(−1)n+1ln(2n−1)(2n−1)
There are various ways to find the derivative, one of the most straigforward is using the Kummer's Fourier series for the log-gamma function:
lnΓ(t)=12ln(πsinπt)+[γ+ln(2π)](12−t)+1π∞∑n=1lnnnsin(2πnt)
It turned out that if t=14 then
sin(πn2)={sin(π2)=1 if n=1sin(π)=0 if n=2sin(3π2)=−1 if n=3sin(2π)=0 if n=4⋮⋮}
Then ∞∑n=1lnnnsin(πn2)=∞∑n=1(−1)n+1ln(2n−1)(2n−1)
Therefore,
∞∑n=1(−1)n+1ln(2n−1)(2n−1)=π[lnΓ(14)−12ln(πsinπ4)−[γ+ln(2π)](14)]=π[ln(π)−12ln(2)−lnΓ(34)−12ln(π)−14ln(2)−14γ−ln(2π)4]=π4[ln(π)−γ−4Γ(34)]
Therefore
13√35√57√79√911√1113√1315√15…=(πeγΓ(34)4)π4
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