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For $|z| < 1$, is the Gaussian hypergeometric function $_{2}F_1(a,b;c;z)$ continuous in the arguments $a$, $b$, and $c$? How to prove it?

Huihui Li
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1 Answers1

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For this hypergeometric function to exist at all, we require that $c$ is not a nonpositive integer ($c \notin -\mathbb N$). Then using the Ratio Test, the series converges absolutely for $|z| < 1$, and convergence is uniform on any compact subset of $\{(a,b,c,z) \in \mathbb C^4: c \notin -\mathbb N, |z| < 1\}$. A uniform limit of continuous functions on a compact set is continuous.

Robert Israel
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  • How do you conclude that absolute convergence implies uniform convergence on compacts in this case? Also, as far as I can tell, Weierstrass test here does not work, right? – Mr_3_7 Oct 18 '22 at 11:04
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    Absolute convergence does not imply uniform convergence, but the estimates from the Ratio Test are uniform on compact sets. – Robert Israel Oct 18 '22 at 23:59