Higher-order asymptotics of generalized hypergeometric function

Question

I have a question about higher-order asymptotics of generalized hypergeometric functions. According to https://dlmf.nist.gov/15.4 the following is well known: $$ _2F_1(a,b;a+b;z)\sim -\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\log(1-z),\ \ z\rightarrow1^{-}. $$ My collaborator was able to coax Wolfram Mathematica into giving a higher-order estimate $$ _2F_1(a,b;a+b;z)\sim -\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\left[\log(1-z)+\psi(a)+\psi(b)+2\gamma\right]+\mathcal{O}((1-z)\log(1-z)) $$ as $z\rightarrow 1^{-}$ for $a$ and $b$ real and positive and where $\psi(z)$ is the digamma function and $\gamma$ is the Euler–Mascheroni constant.

Can anybody provide me with any reference or a hint of why the above Mathematica estimate would be correct?

Answer 1

In Abramowitz and Stegun, Formula 15.3.11, the equation reads for $m=0,$ $${}_2F_1(a,b,a+b) = -\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \sum_{n=0}^\infty \frac{(a)_n (b)_n}{n!^2}(1-z)^n \Big( $$ $$\log(1-z) -2\psi(n+1) + \psi(a+n) + \psi(b+n) \Big)$$ Your asymptotic approximation is the $n=0$ term.