Asymptotic expansion of Hypergeometric2F1[1, m, 1 + m, 2] as $m$ goes to infinity.

Question

I would like to get an asymptotic expansion of $$ \, _2F_1(1,m;m+1;2) $$ as $m \to \infty$.

I have checked DLMF (https://dlmf.nist.gov/15.12), but I have not found anything that may be used for this case.

Answer 1

If your function is defined by the standard reflection formula, then $$F(m) = {_2F_1}(1, m; m + 1; 2) =\\ (-2)^{-m} \pi m \csc \pi m - \frac m {2(m - 1)} \,{_2F_1} \left( 1, 1 - m; 2 - m; \frac 1 2 \right) = \\ (-2)^{-m} \pi m \csc \pi m - \sum_{k = 0}^\infty \frac 1 {1 - \frac {k + 1} m} 2^{-k - 1}.$$ Let $m$ be bounded away from integers by some fixed distance. Then the first term is exponentially small and can be discarded.

The summand will decrease exponentially in $k$, with $(1 - (k + 1)/m)^{-1}$ staying bounded. Then we can expand the summand into a series in $m$ to obtain a complete asymptotic series for $F(m)$: $$F(m) \sim -\sum_{k = 0}^\infty \operatorname{Li}_{-k} \left( \frac 1 2 \right) m^{-k} = -1 - \frac 2 m - \frac 6 {m^2} - \dots \,.$$ $F(m)$ is, in fact, continuous, the integral values of $m$ are removable singularities.