A001044: Beta function proof of Sum a(n)/(2n+1)! = 2*Pi/sqrt(27)

[White Board]

Hi all,

I was playing around with the Beta function recently and ended up with a derivation of Sum_{n>=0} (n!)^2/(2n+1)! = 2*Pi/(3*sqrt(3)) = 2*Pi/sqrt(27) that goes through the Beta function. Thought I'd share it here.

The Beta function is: B(p,q) = Integral_{0..1} t^(p-1) (1-t)^(q-1) dt = Gamma(p) Gamma(q) / Gamma(p+q)

For positive integers p, q this becomes: B(p,q) = (p-1)! (q-1)! / (p+q-1)!
Setting q = p gives, B(p,p) = ((p-1)!)^2 / (2p-1)!

So the sum I'm interested in is: S = Sum_{p>=1} B(p,p)

Using the integral representation and summing under the integral sign (after the substitution t = sin^2(theta) the series converges uniformly on [0,1]). Therefore,

S = Integral_{0..1} Sum_{p>=1} t^(p-1) (1-t)^(p-1) dt
= Integral_{0..1} 1 / (1 - t(1-t)) dt

Since 1 - t(1-t) = t^2 - t + 1, Thus S = Integral_{0..1} dt / (t^2 - t + 1)
= Integral_{0..1} dt / ((t - 1/2)^2 + 3/4)

Substituting u = (2*t - 1)/sqrt(3), it becomes: S = (2/sqrt(3)) * [arctan(u)] from -1/sqrt(3) to 1/sqrt(3)
= (2/sqrt(3)) * (Pi/6 + Pi/6)
= 2*Pi / (3*sqrt(3))
which is the same as 2*Pi/sqrt(27) listed in A001044's formula by Daniel Suteu, Feb 06 2017. As far as I can tell, Suteu's entry gives only the result, without a derivation, and the other formulas in the entry go through Bessel, arcsin and hypergeometric routes-- not Beta. So the proof above seems to be new even though the result isn't. I think it might be worth adding the following comment to A001044:

"Sum_{n>=0} A001044(n)/(2n+1)! = 2*Pi/(3*sqrt(3)) = 2*Pi/sqrt(27)
(proposed by Daniel Suteu, Feb 06 2017) is derived here via the Beta function:
B(p,p) = ((p-1)!)^2/(2p-1)! and
Sum_{p>=1} B(p,p) = Integral_{0..1} dt/(t^2 - t + 1) = 2*Pi/(3*sqrt(3))."

Two small things worth noting:
1. If the sum is indexed from 0 instead of 1, the value is the same, since the p = 0 term vanishes in the integral form. Both conventions give the same constant.
2. With q = p - 1 one gets Sum_{p>=2} (p-1)!(p-2)!/(2p-2)! = Pi/(3*sqrt(3)), exactly half of the above. That might also be worth a cross-reference.

If anyone has seen this derivation before, or knows a reference, please let me know. If the editors are happy with it, I'll submit the comment to A001044.

Regards, Rad

[Stephen Lucas]

Rad, a nice straightforward derivation.

For what it is worth, sequence A073010 is the decimal expansion of pi/sqrt(27), and in the formulas section it includes:

Pi/sqrt(27) = Sum_{n >= 1} 1/(n*binomial(2*n,n)) = (1/6)*Sum_{n >= 1} 3^n/(n*binomial(2*n,n)) (see Lehmer, equation 12, and also p. 456). The Lehmer reference is D. H. Lehmer, Interesting Series Involving the Central Binomial Coefficient, Am. Math. Monthly 92 (1985) 449-457. See eq. (12).

This looks really similar to your formula except for the factor of 2. 

The Lehmer paper is available on JSTOR https://www.jstor.org/stable/2322496?seq=6 and it looks like it is free. I’d check that paper before making changes to A001044.

Steve

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Stephen Lucas, Professor
Department of Mathematics and Statistics
MSC 1911, James Madison University, Harrisonburg, VA 22807 USA
Phone 540 568 5104, Fax 540 568 6857, Web http://educ.jmu.edu/~lucassk/
Email lucassk at jmu dot edu (Work) stephen.k.lucas at gmail dot com (Other)

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