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\begin_body

\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\ud}{\mathrm{d}}
\end_inset


\begin_inset Formula 
\begin{equation}
\Delta_{n}(x,z)\equiv\int_{x}^{\infty}t^{-\frac{1}{2}-n}e^{-t+\frac{z^{2}}{4t}}\ud t\label{eq:Delta definition}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Integration per partes:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\int t^{-\frac{1}{2}-n}\ud t=\frac{t^{\frac{1}{2}-n}}{\frac{1}{2}-n};
\]

\end_inset


\begin_inset Formula 
\[
\frac{\ud}{\ud t}e^{-t+\frac{z^{2}}{4t}}=\left(-1-\frac{z^{2}}{4t^{2}}\right)e^{-t+\frac{z^{2}}{4t}}
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align*}
\left(\frac{1}{2}-n\right)\Delta_{n} & =-x^{\frac{1}{2}-n}e^{-x+\frac{z^{2}}{4x}}+\int_{x}^{\infty}t^{\frac{1}{2}-n}e^{-t+\frac{z^{2}}{4t}}\ud t+\frac{z^{2}}{4}\int_{x}^{\infty}t^{\frac{-3}{2}-n}e^{-t+\frac{z^{2}}{4t}}\ud t\\
 & =-x^{\frac{1}{2}-n}e^{-x+\frac{z^{2}}{4x}}+\Delta_{n-1}+\frac{z^{2}}{4}\Delta_{n+1},
\end{align*}

\end_inset


\begin_inset Formula 
\begin{equation}
\Delta_{n+1}=\frac{4}{z^{2}}\left(\left(\frac{1}{2}-n\right)\Delta_{n}-\Delta_{n-1}+x^{\frac{1}{2}-n}e^{-x+\frac{z^{2}}{4x}}\right).\label{eq:Delta recurrence}
\end{equation}

\end_inset

There are obviously wrong signs in Kambe II, (A 3.3).
\end_layout

\begin_layout Standard
Eq.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Delta recurrence"
plural "false"
caps "false"
noprefix "false"

\end_inset

 is obviously unsuitable for numerical computation when 
\begin_inset Formula $z$
\end_inset

approaches 0.
 However, the definition 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Delta definition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 suggests that the function should be analytical around 
\begin_inset Formula $z=0$
\end_inset

.
 If 
\begin_inset Formula $z=0$
\end_inset

, one has (by definition of incomplete Г function)
\begin_inset Formula 
\begin{equation}
\Delta_{n}(x,0)=\Gamma\left(\frac{1}{2}-n,x\right).\label{eq:Delta:z = 0}
\end{equation}

\end_inset

For convenience, label 
\begin_inset Formula $w=z^{2}/4$
\end_inset

 and 
\begin_inset Formula 
\[
\Delta'_{n}\left(x,w\right)\equiv\int_{x}^{\infty}t^{-\frac{1}{2}-n}e^{-t+\frac{w}{t}}\ud t.
\]

\end_inset

 Differentiating by parameter 
\begin_inset Formula $w$
\end_inset

 (which should be fine as long as the integration contour does not go through
 zero) gives
\begin_inset Formula 
\[
\frac{\partial\Delta'_{n}\left(x,w\right)}{\partial w}=\Delta'_{n+1}\left(x,w\right),
\]

\end_inset

so by recurrence
\begin_inset Formula 
\[
\frac{\partial^{k}}{\partial w^{k}}\Delta'_{n}\left(x,w\right)=\Delta'_{n+k}\left(x,w\right).
\]

\end_inset

Together with 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Delta:z = 0"
plural "false"
caps "false"
noprefix "false"

\end_inset

, this gives an expansion around 
\begin_inset Formula $w=0$
\end_inset

:
\begin_inset Formula 
\[
\Delta_{n}'\left(x,w\right)=\sum_{k=0}^{\infty}\Gamma\left(\frac{1}{2}-n-k,x\right)\frac{w^{k}}{k!},
\]

\end_inset


\begin_inset Formula 
\[
\Delta_{n}\left(x,z\right)=\sum_{k=0}^{\infty}\Gamma\left(\frac{1}{2}-n-k,x\right)\frac{\left(z/2\right)^{2k}}{k!}.
\]

\end_inset

The big negative first arguments in incomplete 
\begin_inset Formula $\Gamma$
\end_inset

 functions should be good (at least I think so, CHECKME), as well as the
 
\begin_inset Formula $1/k!$
\end_inset

 factor (of course).
 I am not sure what the convergence radius is, but for 
\begin_inset Formula $\left|z\right|<2$
\end_inset

 there seems to be absolutely no problem in using this formula.
\end_layout

\end_body
\end_document