**Reference request.** A prototype case:


In
$$
{}\_2F\_1\left(\frac{1}{12},\frac{5}{12};\frac{1}{2};x\right) =
A\log\left(\frac{1}{1-x}\right) + B + o(1),
\qquad x \to 1^-
$$
what can we say about the connection coefficient $B \approx 0.995$? Of course already Gauss knew
$$
A = \frac{\Gamma\left(\frac{7}{12}\right)\Gamma\left(\frac{11}{12}\right)}{4\pi^{3/2}} .
$$
Also well-known are non-log cases of ${}\_2F\_1(a,b;c;x)$, such as those where $a+b-c \notin \mathbb Z$.



hypergeometric at nearest singularity
The result for ${}\_3F\_2$ was suspected by Ramanujan, and proved by Evans and Stanton (Thm 3). <http://epubs.siam.org/doi/abs/10.1137/0515078> <http://www.math.ucsd.edu/~revans/Stanton.pdf>


The general ${}\_3F\_2$ result is:
$${\Gamma(a)\Gamma(b)\Gamma(c)\over\Gamma(d)\Gamma(e)}{}\_3F\_2({a,b,c\atop d,e};z)\sim$$
$$\sim-\log(1-z)-2\gamma-{\Gamma'(a)\over\Gamma(a)}-{\Gamma'(b)\over\Gamma(b)}+\sum\_{k=1}^\infty {(d-c)\_k(e-c)\_k\over k(a)\_k(b)\_k}$$
when $a+b+c=d+e$ and $\Re(c)>0$.


Transferring to ${}\_2F\_1$, in the specified example one has that $B/A$ is
$$-{\Gamma'\over\Gamma}(1/12)-{\Gamma'\over\Gamma}(5/12)-2\gamma.$$


The above paper mentions its equation (1.10) being a "well-known" asymptotic formula for the $m$th partial sum of a zero-balanced ${}\_2F\_1$ (citing Luke p109 #34), and taking the limit as $m\rightarrow\infty$ should already give the desired result for ${}\_2F\_1$.


