# Searching for hypergeometric motives that split

Motivation: It seems that the splitting of a hypergeometric motive is closely related to some highly non-trivial hypergeometric identities discovered by Ramanujan, Guillera et al. The splitting of hypergeometric motives can be treated as a conceptual idea for discovering other Ramanujan-type formulas, such as (using notations from Magma Handbook)

$$H([3,4,6],[2,1,1,1,1,1],729/4)$$

and

$$H([2,3,3],[1,1,1,6],-4)$$

give

$$$$\sum_{k=0}^{\infty}\frac{(1/3)_k(2/3)_k(1/4)_k(3/4)_k(1/6)_k(5/6)_k}{(1)_k^5(1/2)_k}\frac{41760k^3+28512k^2+4264k+220}{2k+1}\left(\frac{4}{729}\right)^k=\frac{2187}{\pi^2}$$$$

and

$$$$\sum_{k=0}^{\infty}\frac{(1/2)_k(1/3)_k^2(2/3)_k^2}{(1)_k^3(1/6)_k(5/6)_k}\frac{270k^3+333k^2+102k+10}{(k+1/6)(k+5/6)}\left(-\frac{1}{4}\right)^k=\frac{108\sqrt{3}}{\pi}$$.$$

(of course, one can obtain their supercongruence counterparts as well.)

Question: We notice that the rational arguments $$t$$ attached to the hypergeometric motives $$H(\alpha,\beta,t)$$ of Ramanujan-type formulas share some interesting properties. In other words, the prime factors of the denominator and the numerator of $$t$$ are quite small.

Examples:

$$H([2,4],[1,1,1],99^4)$$

and

$$H([2,2,2,2,5],[1,1,1,1,1,1,1,1,1],-3125/1024)$$

are hypergeometric motives that (highly likely) split. Even the formula (31) and (38) in Guillera's paper from quadratic fields seem to fit well with the observation.

1. It seems that one can usually succeed in finding out a hypergeometric motive that splits with such $$t$$. Why is it so?
2. Is there any known algorithm for locating an algebraic $$t$$ that $$H(\alpha,\beta,t)$$ splits?
• Sagemath also has basic functionality on hypergeometric motives. – F. C. Apr 24 '19 at 6:42