Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 14E / c / Solution

First assume $a-b\notin \mathbb{Z}$ and nonsingular parameters, so the two infinity solutions $w_1$ and $w_2$, obtained by exchanging $a,b$, are independent. Write $F=A{w}_1+B{w}_2$. Since the power series at $1/z=0$ starts at one, $w_1\sim (-z{)}^{-a}$ and $w_2\sim (-z{)}^{-b}$.

In the parameter region $\Re (b-a)>0$, let $z\to -\infty$ in the Pfaff transformation. The $w_2$ term vanishes after multiplication by $(-z{)}^a$, while $z/(z-1)\to 1$ and $(-z)/(1-z)\to 1$. The supplied value at one of the Gauss hypergeometric function therefore givesAnalytic continuation in the parameters extends this coefficient to the nonresonant parameter domain. Exchange $a,b$ to determine $B$. Hence the hypergeometric connection formula at infinity isThe branch is $|\arg (-z)|<\pi$. The existence of $F(a,b;c;1)$ alone does not exclude integer $a-b$. At such resonant values the two separate displayed terms can be singular; the correct statement takes their combined parameter limit, which can contain logarithms. This qualification is essential, for example when $a=b$.