WM Whittaker M

WM.1 Introduction

Let x be a complex variable of ℂ\{0, ∞} and let μ, ν denote a set of parameters (independent of x).The function Whittaker M (noted WM_{μ,ν}) is defined by the following second order differential equation

\begin{aligned}& −x^{2} − 4μx − 1 + 4ν^{2}y(x) + 4x^{2} \frac{∂^{2}y(x)}{∂x^{2}} & = 0. \\ (WM.1.1) \end{aligned}

Although 0 is a singularity of WM.1.1, the initial conditions can be given by

\begin{aligned}& \frac{∂_{\frac{WM_{μ,ν}(x)}{x^{(ν+ \frac{1}{2})}}}}{∂x} & = 1. \\ (WM.1.2) \end{aligned}

The formulae of this document are valid for 2ν ∉ ℤ. Related function: Whittaker W

WM.2 Series and asymptotic expansions

WM.2.1 Asymptotic expansion at 0.

WM.2.1.1 First terms.

\begin{matrix}WM_{μ,ν}(x) ≈ x^{(ν+ \frac{1}{2})}(1 − \frac{μx}{2ν + 1} + \frac{(2ν + 1 + 4μ^{2})x^{2}}{(8ν + 8)(4ν + 2)} + \\ \frac{−μ(6ν + 5 + 4μ^{2})x^{3}}{12(8ν + 12)(2ν + 1)(ν + 1)}+ \\ \frac{(12ν^{2} + 24ν + 9 + 48μ^{2}ν + 56μ^{2} + 16μ^{4})x^{4}}{192(8ν + 16)(2ν + 1)(ν + 1)(2ν + 3)} + \\ \frac{−μ(60ν^{2} + 160ν + 89 + 80μ^{2}ν + 120μ^{2} + 16μ^{4})x^{5}}{1920(8ν + 20)(2ν + 1)(ν + 1)(2ν + 3)(ν + 2)} + (( \\ 120ν^{3} + 540ν^{2} + 690ν + 225 + 720μ^{2}ν^{2} + 2400μ^{2}ν + 1756μ^{2} \\ + 480μ^{4}ν + 880μ^{4} + 64μ^{6})x^{6})/ \\ (46080(8ν + 24)(2ν + 1)(ν + 1)(2ν + 3)(ν + 2)(2ν + 5)) + (−μ( \\ 840ν^{3} + 4620ν^{2} + 7518ν + 3429 + 1680μ^{2}ν^{2} + 6720μ^{2}ν+ \\ 6076μ^{2} + 672μ^{4}ν + 1456μ^{4} + 64μ^{6})x^{7})/(645120(8ν + 28) \\ (2ν + 1)(ν + 1)(2ν + 3)(ν + 2)(2ν + 5)(ν + 3)) …). \end{matrix} ; (WM.2.1.1.1)

WM.2.1.2 General form.

\begin{aligned}& WM_{μ,ν}(x) ≈ x^{(ν+ \frac{1}{2})} ∑_{n=0}^{∞} u(n)x^{n}. \\ (WM.2.1.2.1) \end{aligned}

The coefficients u(n) satisfy the recurrence

\begin{matrix}u(n)(4(ν + \frac{1}{2} + n)^{2} − 4ν − 1 − 4n − 4ν^{2}) + 4u(n − 1)μ − u(n − 2) = 0. \\ (WM.2.1.2.2) \end{matrix}

Initial conditions of WM.2.1.2.2 are given by

\begin{aligned}u(1) & = \frac{−4μ}{8ν + 4}, \\ u(0) & = 1. \end{aligned} ; (WM.2.1.2.3)

The recurrence WM.2.1.2.2 has the closed form solution

\begin{aligned}& u(n) & = 0. \\ (WM.2.1.2.4) \end{aligned}

WM.2.2 Asymptotic expansion at ∞. WM.2.2.1 First terms.

WM_{μ,ν}(x) ≈ e^{\frac{1}{2x}} x^{μ}y_{0}(x), where

y_{0}(x) = \frac{Γ(2ν + 1)}{Γ(\frac{1}{2} + ν − μ)} + \frac{−(4ν^{2} − 4μ − 4μ^{2} − 1)Γ(2ν + 1)x}{4Γ(\frac{1}{2} + ν − μ)}+

\frac{(4ν^{2} − 12μ − 4μ^{2} − 9)(4ν^{2} − 4μ − 4μ^{2} − 1)Γ(2ν + 1)x^{2}}{32Γ(\frac{1}{2} + ν − μ)}

+ (−(4ν^{2} − 20μ − 4μ^{2} − 25)(4ν^{2} − 12μ − 4μ^{2} − 9)

(4ν^{2} − 4μ − 4μ^{2} − 1)Γ(2ν + 1)x^{3})/(384Γ(\frac{1}{2} + ν − μ)) + 2 …

WM.2.2.2 General form.

WM.2.2.2.1 Auxiliary function y_{0}(x). The coefficients u(n) of y_{0}(x) satisfy the following recurrence

\begin{matrix}− 4u(n)n+ \\ u(n − 1)(−4ν^{2} + 4μ + 4μ^{2} − 3 + 4n + 8(n − 1)μ + 4(n − 1)^{2}) = 0 \end{matrix}

whose initial conditions are given by

u(0) = \frac{Γ(2ν + 1)}{Γ(\frac{1}{2} + ν − μ)}

This recurrence has the closed form solution

u(n) = (sin(\frac{π(1 + 2ν − 2μ)}{2})(−2)^{n}2^{n}(−1)^{n}Γ(n − ν + \frac{1}{2} + μ)

Γ(n + ν + \frac{1}{2} + μ)Γ(2ν + 1))/(4^{n}πΓ(ν + \frac{1}{2} + μ)Γ(n + 1)).