I. Define the ff integrals,

$$K(k)=K_2(k)=\int_0^{\pi/2}\frac{1}{\sqrt{1-k^2 \sin^2 x}}dx=\large{\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$

$$K_3(k)=\int_0^{\pi/2}\frac{\cos\left(\frac13\,\arcsin\big(k\sin x\big)\right)}{\sqrt{1-k^2 \sin^2 x}}dx=\large{\tfrac{\pi}{2}\,_2F_1\left(\tfrac13,\tfrac23,1,\,k^2\right)}$$

$$K_4(k)=\int_0^{\pi/2}\frac{\cos\left(\frac12\,\arcsin\big(k\sin x\big)\right)}{\sqrt{1-k^2 \sin^2 x}}dx=\large{\tfrac{\pi}{2}\,_2F_1\left(\tfrac14,\tfrac34,1,\,k^2\right)}$$

$$K_6(k)=\int_0^{\pi/2}\frac{\cos\left(\frac23\,\arcsin\big(k\sin x\big)\right)}{\sqrt{1-k^2 \sin^2 x}}dx=\large{\tfrac{\pi}{2}\,_2F_1\left(\tfrac16,\tfrac56,1,\,k^2\right)}$$

These are Ramanujan's theory of elliptic functions for alternative bases of signature $2,3,4,6$, respectively. There are only 4 signatures.

II. Then, using Wolfram, I observed the closed-forms of the ff definite integrals,

$$\int_0^1 K_2(k)\, dk = {\tfrac{\pi}{2}\,_3F_2\left(\tfrac12,\tfrac12,\tfrac12;1,\tfrac32;1\right)}=2G$$

$$\int_0^1 K_3(k)\, dk = {\tfrac{\pi}{2}\,_3F_2\left(\tfrac12,\tfrac13,\tfrac23;1,\tfrac32;1\right)}=\tfrac{3\sqrt3}2\, \ln2$$

$$\int_0^1 K_4(k)\, dk = {\tfrac{\pi}{2}\,_3F_2\left(\tfrac12,\tfrac14,\tfrac34;1,\tfrac32;1\right)}=2\ln(1+\sqrt2)$$

$$\int_0^1 K_6(k)\, dk = {\tfrac{\pi}{2}\,_3F_2\left(\tfrac12,\tfrac16,\tfrac56;1,\tfrac32;1\right)}=\tfrac{3\sqrt3}4\, \ln(2+\sqrt{3})$$

where $G$ is Catalan's constant. (Curiously, other than the first, Wolfram didn't recognize the closed-form of those hypergeometrics. I had to use the Inverse Symbolic Calculator.)

III. Questions

  1. Does the generalized hypergeometric function, $$H(n)=\,_3F_2\left(\tfrac12,\tfrac1n,\tfrac{n-1}{n};1,\tfrac32;1\right)$$ have a closed form only for $n=2,3,4,6$? (I tried $n=5,7,8$, etc, and it doesn't seem to have a "neat" form using elementary functions.)
  2. If so, is it connected to why there are only 4 signatures of alternative bases?
  • $\begingroup$ $H(1)=\frac{\pi}2.$ $\endgroup$ – Somos Jan 29 '19 at 19:58
  • $\begingroup$ Fascinating! I wish I could help :) $\endgroup$ – clathratus Jan 30 '19 at 2:16
  • $\begingroup$ The numerators of the $2G$ series is given here oeis.org/A038534 In Mathermatica given by $$\binom{2 n}{n}^2 2^{-2\, \text{DigitCount}[n,2,1]}$$ The DigitCount function giving the number of 1's in a base 2 number $n$. Wild! $\endgroup$ – James Arathoon Jan 30 '19 at 13:08
  • $\begingroup$ Nice integral representations for alternative $K$s. +1 $\endgroup$ – Paramanand Singh Jan 31 '19 at 5:19
  • $\begingroup$ @ParamanandSingh: I believe it's the general form. The signature $2$ numerator only looks different since $a-b = \frac12-\frac12 = 0$ and $\cos(0\,\alpha) = 1$. $\endgroup$ – Tito Piezas III Jan 31 '19 at 6:07

Some speculative clues have appeared after a bit of detective work...

First the integral we are interested in with the associated hypergeometric function and infinite series.

$$I_n=\int_0^1 K_n(k)\, dk = {\tfrac{\pi}{2}\,_3F_2\left(\tfrac12,\tfrac{1}{n},\tfrac{n-1}{n};1,\tfrac32;1\right)}= \frac{ \pi}{2}\times\sum _{k=0}^{\infty } \frac{\prod _{j=0}^{k-1} \left(j+\frac{1}{n}\right) \prod _{j=0}^{k-1} \left(j+\frac{n-1}{n}\right)}{(2 k+1) (k!)^2}$$

Simplifying the infinite series a little I found that $$I_n=\frac{ \pi}{2}\,\sum _{k=0}^{\infty } \frac{\prod _{j=1}^k \left(j^2-\frac{1}{n^2}\right)}{(k n+1)(2 k+1) (k!)^2 }$$

Now some interesting links appear to your integral if we study the much simpler sum

$$S_n=\sum _{k=0}^{\infty } \frac{(-1)^k}{(k n+1)( 2k+1)}$$

we find from Mathematica that $$S_2=G$$ $$S_3=\pi \left(\frac{1}{\sqrt{3}}-\frac{1}{2}\right)+\log (2)$$ $$S_4=\frac{1}{4} \pi \left(\sqrt{2}-1\right)+\frac{\log \left(\sqrt{2}+1\right)}{\sqrt{2}}$$ $$S_6=\frac{1}{8} \left(\pi +2 \sqrt{3} \log \left(\sqrt{3}+2\right)\right)$$ $$S_8=\frac{1}{12} \pi \left(\sqrt{2}+1\right)+\frac{\log (2)}{3}+\frac{\log \left(\sqrt{2}+1\right)}{3 \sqrt{2}}$$

These are all the shortest and simplest closed forms between $n=2$ and $n=12$.

For $I_2$, $I_3$, $I_4$ and $I_6$ that you found closed forms for, the respective sums have one term with the same principal constant and have a maximum of 3 terms. The next simplest sum I found is $S_8$ with four terms.

Have fun.

  • $\begingroup$ Interesting. I tried using an integer relations program on the 6 vectors $I_8, \sqrt2,\ln2,\sqrt2\ln2,\ln(1+\sqrt2),\sqrt2\ln(1+\sqrt2)$ assuming $I_8$ is a rational sum involving those. Unfortunately, it couldn't find a simple relation. $\endgroup$ – Tito Piezas III Jan 31 '19 at 3:07
  • $\begingroup$ By the way, I fixed a small typo in your simplified series for $I_n$. There should be a $\pi/2$. $\endgroup$ – Tito Piezas III Jan 31 '19 at 3:20
  • $\begingroup$ Thanks for fixing that. $\endgroup$ – James Arathoon Jan 31 '19 at 8:40

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