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The following classical identity is well-known, $$_2F_1(a,\, 1 - a;\, -a;\, z) = \frac{1 - 2 z}{(1 - z)^{1 + a}}$$

Q1: Let $a,b,c$ be rationals. Given $_2F_1(a, b; c; z)$ with non-constant $a$ and non-constant algebraic number $z$, what other broad families are there such that $_2F_1$ also is an algebraic number?

Doing some computer searches complemented by the DLMF and others, I found the following tentative list. Let $m,n$ be any integer:

$$_2F_1(a,\, b;\, a-n;\, z)\tag1$$ $$_2F_1(a,\, 1-a;\, \pm a-n;\, z)\tag2$$ $$_2F_1(a,\, \pm m-a;\, \tfrac12\pm n;\, z)\tag3$$ $$_2F_1(a,\, \pm\tfrac12 +a;\, \tfrac12\pm n;\, z)\tag4$$ $$_2F_1(a,\, \pm\tfrac12 +a;\, 2a\pm n;\, z)\tag5$$

Thus the example was just $(2)$ with $n=0$. (I didn't include the general evaluation as they seem to be complicated.)

Q2: What in the list should be trimmed or broaden? And what family can be added?

P.S. The family must not have a constant $a$ nor $z$.

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    @Nemo: How can family $(3)$ be reducible to just two parameters? – 2017-01-07
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    @Nemo: That is family $(5)$ with your $n=2a$. – 2017-01-07
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    I think yes, the integer parameter $n$ in (5) is superfluous. The main identity is the one with $n=0$ and the case $n\in\mathbb{N}$ is obtained by differentiation. – 2017-01-07
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    @Nemo: I see. However, if you can derive the other four families from $(5)$, I'll be happy to accept that as an answer. :) – 2017-01-07
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    I think I was wrong. (3) and (5) with $m=n=0$ are main identities, and (4) is consequence of (3). (2) is consequnce of (1) and (1) is consequence of http://dlmf.nist.gov/15.4.E6 – 2017-01-07

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