**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 $a+b-c \notin \mathbb Z$.