Darboux's method

If we have a function ${\displaystyle (1-z)^{\beta }f(z)}$ where ${\displaystyle \beta \notin \{0,1,2,\ldots \}}$ and ${\displaystyle f(z)}$ has a radius of convergence greater than ${\displaystyle 1}$ and a Taylor expansion near 1 of ${\displaystyle \sum _{j\geq 0}f_{j}(1-z)^{j}}$, then^[17]

${\displaystyle [z^{n}](1-z)^{\beta }f(z)=\sum _{j=0}^{m}f_{j}{\frac {n^{-\beta -j-1}}{\Gamma (-\beta -j)}}+O(n^{-m-\beta -2})}$

See Szegő (1975) for a similar theorem dealing with multiple singularities.

Singularity analysis

If ${\displaystyle f(z)}$ has a singularity at ${\displaystyle \zeta }$ and

${\displaystyle f(z)\sim \left(1-{\frac {z}{\zeta }}\right)^{\alpha }\left({\frac {1}{\frac {z}{\zeta }}}\log {\frac {1}{1-{\frac {z}{\zeta }}}}\right)^{\gamma }\left({\frac {1}{\frac {z}{\zeta }}}\log \left({\frac {1}{\frac {z}{\zeta }}}\log {\frac {1}{1-{\frac {z}{\zeta }}}}\right)\right)^{\delta }\quad }$ as ${\displaystyle z\to \zeta }$

where ${\displaystyle \alpha \notin \{0,1,2,\cdots \},\gamma ,\delta \notin \{1,2,\cdots \}}$ then^[18]

${\displaystyle [z^{n}]f(z)\sim \zeta ^{-n}{\frac {n^{-\alpha -1}}{\Gamma (-\alpha )}}(\log n)^{\gamma }(\log \log n)^{\delta }\quad }$ as ${\displaystyle n\to \infty }$