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I suspect that the Gauss hypergeometric function ${}_{2}F_{1}(a,b,c, m z)$ can be rewritten as a function of ${}_{2}F_{1}(a,b,c, z)$, with $m \in \mathbb{N}$. More formally, I need to find the function $f$ such that

$$f \big({}_{2}F_{1}(a,b,c,z),m\big)={}_{2}F_{1}(a,b,c, m z).$$

However, I couldn't prove such property so far. Is there any way to rewrite the hypergeometric function when the argument $z$ is multiplied by a positive integer $m$?

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    Have you tried the series definition of the Gauss function? Or the integral definition? Mind, it's not clear to me if it's possible to obtain such a function – 2017-01-16
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    By the way, what is your suspicion based on? – 2017-01-16

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