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\begin{document}
\chapter*{WM Whittaker M}
\label{WM}
\section*{WM.1 Introduction}
\label{WM:intro}


Let $x$ be a complex variable of $\mathbb{C} \setminus \{0,\infty\}$ and let $\mu,\nu$ denote a set of parameters (independent of $x$).The function Whittaker M (noted $\operatorname{WM} _{\mu , \nu}$) is defined by the following second order differential equation
\begin{equation*}
\label{WM:diffeq}
\begin{split}
-x^{2} - 4 \mu x - 1 + 4 \nu^{2}y (x) + 4 x^{2} \frac{\partial^{2} y (x)}{\partial x^{2}}& =0.
\end{split}\tag{WM.1.1}
\end{equation*}



Although $0$ is a singularity of WM.1.1, the initial conditions can be given by 
\begin{equation*}
\label{WM:inicond}
\begin{split}
\frac{\partial \frac{\operatorname{WM} _{\mu , \nu} (x)}{x^{\bigl(\nu + \frac{1}{2}\bigr)}}}{\partial x}& =1.
\end{split}\tag{WM.1.2}
\end{equation*}



The formulae of this document are valid for $2 \nu \not\in \mathbb{Z} .$


Related function: \href{http://algo.inria.fr/esf/function/WW/WW.html#WW}{Whittaker W}
\section*{WM.2 Series and asymptotic expansions}
\label{WM:asympt}
\subsection*{WM.2.1 Asymptotic expansion at $0$}
\label{743657673814948801}
\subsubsection*{WM.2.1.1 First terms}
\label{WM:asympt:0:termsec}



\begin{equation*}
\label{WM:asympt:0:terms}
\begin{split}
& \operatorname{WM} _{\mu , \nu} (x)\approx x^{\Bigl(\nu + \frac{1}{2}\Bigr)} \Biggl(1 - \frac{\mu x}{2 \nu + 1} + \frac{\bigl(2 \nu + 1 + 4 \mu^{2}\bigr) x^{2}}{(8 \nu + 8) (4 \nu + 2)} +  \\
& \quad{}\quad{}\frac{-\mu \bigl(6 \nu + 5 + 4 \mu^{2}\bigr) x^{3}}{12 (8 \nu + 12) (2 \nu + 1) (\nu + 1)} +  \\
& \quad{}\quad{}\frac{\bigl(12 \nu^{2} + 24 \nu + 9 + 48 \mu^{2} \nu + 56 \mu^{2} + 16 \mu^{4}\bigr) x^{4}}{192 (8 \nu + 16) (2 \nu + 1) (\nu + 1) (2 \nu + 3)} +  \\
& \quad{}\quad{}\frac{-\mu \bigl(60 \nu^{2} + 160 \nu + 89 + 80 \mu^{2} \nu + 120 \mu^{2} + 16 \mu^{4}\bigr) x^{5}}{1920 (8 \nu + 20) (2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2)} + \Bigl(\bigl( \\
& \quad{}\quad{}120 \nu^{3} + 540 \nu^{2} + 690 \nu + 225 + 720 \mu^{2} \nu^{2} + 2400 \mu^{2} \nu + 1756 \mu^{2}  \\
& \quad{}\quad{}+ 480 \mu^{4} \nu + 880 \mu^{4} + 64 \mu^{6}\bigr) x^{6}\Bigr)\bigg/ \\
& \quad{}\quad{}\bigl(46080 (8 \nu + 24) (2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2) (2 \nu + 5)\bigr) + \Bigl(-\mu \bigl( \\
& \quad{}\quad{}840 \nu^{3} + 4620 \nu^{2} + 7518 \nu + 3429 + 1680 \mu^{2} \nu^{2} + 6720 \mu^{2} \nu +  \\
& \quad{}\quad{}6076 \mu^{2} + 672 \mu^{4} \nu + 1456 \mu^{4} + 64 \mu^{6}\bigr) x^{7}\Bigr)\bigg/\bigl(645120 (8 \nu + 28)  \\
& \quad{}\quad{}(2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2) (2 \nu + 5) (\nu + 3)\bigr)\ldots\Biggr).
\end{split}\tag{WM.2.1.1.1}
\end{equation*}

\subsubsection*{WM.2.1.2 General form}
\label{WM:asympt:0:genf}



\begin{equation*}
\label{WM:asympt:0:genfsum}
\begin{split}
& \operatorname{WM} _{\mu , \nu} (x)\approx x^{\Bigl(\nu + \frac{1}{2}\Bigr)} \sum_{n = 0}^{\infty} u (n) x^{n}.
\end{split}\tag{WM.2.1.2.1}
\end{equation*}
The coefficients $u (n)$ satisfy the recurrence
\begin{equation*}
\label{WM:asympt:0:genfrec}
\begin{split}
& u (n) \Biggl(4 \biggl(\nu + \frac{1}{2} + n\biggr)^{2} - 4 \nu - 1 - 4 n - 4 \nu^{2}\Biggr) + 4 u (n - 1) \mu - u (n - 2)=0.
\end{split}\tag{WM.2.1.2.2}
\end{equation*}
Initial conditions of WM.2.1.2.2 are given by
\begin{equation*}
\label{WM:asympt:0:genfic}
\begin{split}
u (1)& =\frac{-4\mu}{8 \nu + 4}, \\
u (0)& =1.
\end{split}\tag{WM.2.1.2.3}
\end{equation*}
The recurrence WM.2.1.2.2 has the closed form solution
\begin{equation*}
\label{WM:asympt:0:RDINREFRDGENFROMRDCLOSED}
\begin{split}
u (n)& =0.
\end{split}\tag{WM.2.1.2.4}
\end{equation*}

\subsection*{WM.2.2 Asymptotic expansion at $\infty$}
\label{743659753679069033}
\subsubsection*{WM.2.2.1 First terms}
\label{743658888210519937}



\begin{equation*}
\begin{split}
& \operatorname{WM} _{\mu , \nu} (x)\approx \operatorname{e} ^{\frac{1}{2 x}} x^{\mu} y _{0} (x),
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
y _{0} (x)& =\frac{\Gamma (2 \nu + 1)}{\Gamma \Bigl(\frac{1}{2} + \nu - \mu\Bigr)} + \frac{-\bigl(4 \nu^{2} - 4 \mu - 4 \mu^{2} - 1\bigr) \Gamma (2 \nu + 1) x}{4 \Gamma \Bigl(\frac{1}{2} + \nu - \mu\Bigr)} +  \\
& \quad{}\quad{}\frac{\bigl(4 \nu^{2} - 12 \mu - 4 \mu^{2} - 9\bigr) \bigl(4 \nu^{2} - 4 \mu - 4 \mu^{2} - 1\bigr) \Gamma (2 \nu + 1) x^{2}}{32 \Gamma \Bigl(\frac{1}{2} + \nu - \mu\Bigr)}  \\
& \quad{}\quad{}+ \Bigl(-\bigl(4 \nu^{2} - 20 \mu - 4 \mu^{2} - 25\bigr) \bigl(4 \nu^{2} - 12 \mu - 4 \mu^{2} - 9\bigr)  \\
& \quad{}\quad{}\bigl(4 \nu^{2} - 4 \mu - 4 \mu^{2} - 1\bigr) \Gamma (2 \nu + 1) x^{3}\Bigr)\Bigg/\Biggl(384 \Gamma \biggl(\frac{1}{2} + \nu - \mu\biggr)\Biggr) + 2 \ldots
\end{split}
\end{equation*}

\subsubsection*{WM.2.2.2 General form}
\label{743659957578069454}
\paragraph*{WM.2.2.2.1 Auxiliary function $y _{0} (x)$}
\label{743659713395048008}
The coefficients $u (n)$ of $y _{0} (x)$ satisfy the following recurrence
\begin{equation*}
\begin{split}
& -4u (n) n +  \\
& u (n - 1) \bigl(-4\nu^{2} + 4 \mu + 4 \mu^{2} - 3 + 4 n + 8 (n - 1) \mu + 4 (n - 1)^{2}\bigr)=0
\end{split}
\end{equation*}
whose initial conditions are given by
\begin{equation*}
\begin{split}
u (0)& =\frac{\Gamma (2 \nu + 1)}{\Gamma \Bigl(\frac{1}{2} + \nu - \mu\Bigr)}
\end{split}
\end{equation*}
This recurrence has the closed form solution
\begin{equation*}
\begin{split}
u (n)& =\Biggl(\operatorname{sin} \Biggl(\frac{\pi (1 + 2 \nu - 2 \mu)}{2}\Biggr) (-2)^{n} 2^{n} (-1)^{n} \Gamma \biggl(n - \nu + \frac{1}{2} + \mu\biggr)  \\
& \quad{}\quad{}\Gamma \biggl(n + \nu + \frac{1}{2} + \mu\biggr) \Gamma (2 \nu + 1)\Biggr)\Bigg/\Biggl(4^{n} \pi \Gamma \biggl(\nu + \frac{1}{2} + \mu\biggr) \Gamma (n + 1)\Biggr).
\end{split}
\end{equation*}
\end{document}