Let us define the complete elliptic integral of the first kind as follows: $$ K(k)=\int_{0}^{1} \frac{1}{\sqrt{1-t^2}\sqrt{1-k^2t^2} } \text{d}t, $$ where we restrict the modulus $k$ to $|k|<1$ with its complementary $K^\prime(k)=K\left ( \sqrt{1-k^2} \right )$. They are shortened as $K,K^\prime$ respectively. Now consider series of $M(p;q,r)=\int_{0}^{1}k^pK^qK^\prime{}^r\text{d}k$ where $p,q,r$ are natural numbers satisfying $p\le q+r-2$, denoting all the moments with $q+r=n$ as the set $I_n$. We have some facts:
- $I_3\in\left \langle M(0;3,0)\right \rangle _{\mathbb{Q}},I_5\in\left \langle M(0;5,0),M(1;5,0)\right \rangle _{\mathbb{Q}}$. $\left \langle \right \rangle _{\mathbb{Q}}$ means the corresponding $\mathbb{Q}$-vector space generated by the selected basis.
- In $I_5$, virtually every element needs two coefficients except the trivial $(0;5,0)=(0;5,0),(1;5,0)=(1;5,0)$ and the remarkable $(0;2,3)=4(1;2,3)=4(1;3,2).$
- For $I_7\in\left \langle M(0;7,0),M(1;7,0),\ \Gamma\left ( \frac14 \right )^{16}/\pi^4 \right \rangle_{\mathbb{Q}}$, we usually see four-term relations like $\int_{0}^{1}(1-k^2-4k^4)K^7\text{d}k= -\frac{63\,\Gamma\left ( \frac14 \right )^{16}}{491520\pi^4}$ but for (they are three-term relations) $$ \begin{aligned} &\int_{0}^{1}kK^3K^\prime\left ( K^3-kK^\prime{}^3 \right ) \text{d} k = \frac{3\,\Gamma\left ( \frac14 \right )^{16} }{229376\pi^4},\\ &\int_{0}^{1}k^3K^3K^\prime\left ( 3K^3-4kK^\prime{}^3 \right ) \text{d} k = \frac{\,\Gamma\left ( \frac14 \right )^{16} }{28672\pi^4},\\ &\int_{0}^{1}k^3K^5K^\prime\left ( 8K^\prime{}-k^2K \right ) \text{d} k = \frac{\,\Gamma\left ( \frac14 \right )^{16} }{21504\pi^4},\\ &\int_{0}^{1}k^3K^4K^\prime{}^2\left ( 15K^\prime{}-8kK \right ) \text{d} k = \frac{\,\Gamma\left ( \frac14 \right )^{16} }{98304\pi^4}. \end{aligned} $$
To explore these relations, one can utilize the property $M\left ( p;q,r \right ) =\int_{0}^{1}k(1-k^2)^{(p-1)/2}K^rK^\prime{^q\text{d}k}$, quadratic transformations (i.e. substituting $k\rightarrow \frac{1-k}{1+k}$) and contour integration (considering functions like $k^pK^qK^\prime{}^r$). And here are my questions:
How many irreducible moments in $I_{2n+1}$ for $n\ge1$? I conjecture it should be $n$.
Do we have any tools to well describe these rational algebraic identities?