I solved an indefinite integral that gave me the following result:

\begin{equation} f(u) = -\sum_{k=0}^{n}\binom{n}{k}\frac{\mathrm{sgn}(u)^{k+1}\, b^{n-k}}{2\, a^{\frac{k+1}{2}}}\, \Gamma\left(\frac{k+1}{2},\, au^{2}\right)+C \end{equation}

where, $u\in (-\infty, \infty)$, $a\geq0$, $b\in (-\infty, \infty)$, and $n$ is a positive integer.

Any ideas of how to even attempt solving this summation? Other related series in this project I am working on have turned up to contain confluent hypergeometric functions if that's a hint. I am also interested in special cases of $u=+\infty$, $\lim_{u\to 0^{+}}$, $\lim_{u\to 0^{-}}$.

Any hints, techniques, or identities would be helpful. Thanks.


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