The Euler $\psi$ function is exactly the same as the digamma function,
$$\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}.$$
(For its use as "Euler $\psi$ function" in the literature, see e.g. this paper.) While the Gamma function, the Pochhammer symbols, and the like, are very useful in *constructing* the hypergeometric functions, it is not possible to express either $\psi$ or $\Gamma$ as special cases of the hypergeometric family.

The digamma function is mildly useful in finding the derivatives of ${}_2F_1(a,b;c;z)$ with respect to the parameters $a,b$ and $c$, though: since
$${}_2F_1(a,b;c;z)=\sum_{n=0}^\infty \frac{\Gamma(a+n)}{\Gamma(a)}\frac{(b)_n}{(c)_n}\frac{z^n}{n!},$$
differentiating by $a$ you can reduce this to
$$\frac{\partial}{\partial a}{}_2F_1(a,b;c;z)
=-\psi(a){}_2F_1(a,b;c;z)
+\sum_{n=0}^\infty \psi(a+n)
\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}.$$
I don't know what good it'll do you, though.

In normal circumstances I would refer you to the Digital Library of Mathematical Functions, which supersedes Abramowitz and Stegun, and particularly chapter 5. For the moment, though, you'll have to make do with the print version.

This post imported from StackExchange Physics at 2014-03-07 13:47 (UCT), posted by SE-user Emilio Pisanty