§13.24 Series

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Keywords:: Whittaker functions, series expansions
Notes:: See Slater (1960, §2.7.3) and Buchholz (1969, §7.4). In the first reference (2.7.16) contains an error.
Referenced by:: §13.11
Permalink:: http://dlmf.nist.gov/13.24
See also:: Annotations for Ch.13

§13.24(i) Expansions in Series of Whittaker Functions

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For expansions of arbitrary functions in series of $M_{\kappa,\mu}\left(z\right)$ functions see Schäfke (1961b).

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Keywords:: Whittaker functions, in Bessel functions or modified Bessel functions, series expansions
Permalink:: http://dlmf.nist.gov/13.24.ii
See also:: Annotations for §13.24 and Ch.13

For $z\in\mathbb{C}$ , and again with the notation of §§10.2(ii) and 10.25(ii),

13.24.1		$M_{\kappa,\mu}\left(z\right)=\Gamma\left(\kappa+\mu\right)2^{2\kappa+2\mu}z^{% \frac{1}{2}-\kappa}\\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(2\kappa+2\mu% \right)_{s}}{\left(2\kappa\right)_{s}}}{{\left(1+2\mu\right)_{s}}s!}\\left(% \kappa+\mu+s\right)I_{\kappa+\mu+s}\left(\tfrac{1}{2}z\right),$
$2\mu,\kappa+\mu\neq-1,-2,-3,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $s$ : nonnegative integer and $z$ : complex variable Referenced by: (13.11.2), §13.24(ii) Permalink: http://dlmf.nist.gov/13.24.E1 Encodings: TeX, pMML, png See also: Annotations for §13.24(ii), §13.24 and Ch.13

and

13.24.2		$\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(z\right)=2^{2\mu}z^{\mu% +\frac{1}{2}}\sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(2\sqrt{\kappa z}\right)^% {-2\mu-s}J_{2\mu+s}\left(2\sqrt{\kappa z}\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $s$ : nonnegative integer, $z$ : complex variable and $p_{s}$ : coefficents Permalink: http://dlmf.nist.gov/13.24.E2 Encodings: TeX, pMML, png See also: Annotations for §13.24(ii), §13.24 and Ch.13

where $p_{0}^{(\mu)}(z)=1$ , $p_{1}^{(\mu)}(z)=\frac{1}{6}z^{2}$ , and higher polynomials $p_{s}^{(\mu)}(z)$ are defined by

13.24.3		$\exp\left(-\tfrac{1}{2}z\left(\coth t-\frac{1}{t}\right)\right)\left(\frac{t}{% \sinh t}\right)^{1-2\mu}=\sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(-\frac{t}{z}% \right)^{s}.$
ⓘ Symbols: $\exp\NVar{z}$ : exponential function, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, $s$ : nonnegative integer, $z$ : complex variable and $p_{s}$ : coefficents Permalink: http://dlmf.nist.gov/13.24.E3 Encodings: TeX, pMML, png See also: Annotations for §13.24(ii), §13.24 and Ch.13

(13.18.8) is a special case of (13.24.1).

Additional expansions in terms of Bessel functions are given in Luke (1959). See also López (1999).

For other series expansions see Prudnikov et al. (1990, §6.6). See also §13.26.