13 Confluent Hypergeometric Functions Whittaker Functions 13.23 Integrals 13.25 Products

§13.24 Series

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

§13.24(i) Expansions in Series of Whittaker Functions
§13.24(ii) Expansions in Series of Bessel Functions

§13.24(i) Expansions in Series of Whittaker Functions

Keywords:: Whittaker functions
Permalink:: http://dlmf.nist.gov/13.24.i

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

§13.24(ii) Expansions in Series of Bessel Functions

Keywords:: Whittaker functions
Permalink:: http://dlmf.nist.gov/13.24.ii

For $z\in\Complex$ , and again with the notation of §§10.2(ii) and 10.25(ii),

13.24.1 $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\mathop{\Gamma\/}% \nolimits\!\left(\kappa+\mu\right)2^{{2\kappa+2\mu}}z^{{\frac{1}{2}-\kappa}}\*% \sum_{{s=0}}^{{\infty}}(-1)^{s}\frac{\left(2\kappa+2\mu\right)_{{s}}\left(2% \kappa\right)_{{s}}}{\left(1+2\mu\right)_{{s}}s!}\*\left(\kappa+\mu+s\right)% \mathop{I_{{\kappa+\mu+s}}\/}\nolimits\!\left(\tfrac{1}{2}z\right),$ $2\mu,\kappa+\mu\neq-1,-2,-3,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : nonnegative integer and : complex variable
Referenced by:: §13.24(ii)
Permalink:: http://dlmf.nist.gov/13.24.E1
Encodings:: TeX, pMML, png

and

13.24.2 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)}\mathop{M_{{\kappa,% \mu}}\/}\nolimits\!\left(z\right)=2^{{2\mu}}z^{{\mu+\frac{1}{2}}}\sum_{{s=0}}^% {{\infty}}p_{s}^{{(\mu)}}(z)\left(2\sqrt{\kappa z}\right)^{{-2\mu-s}}\mathop{J% _{{2\mu+s}}\/}\nolimits\!\left(2\sqrt{\kappa z}\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : nonnegative integer, : complex variable and $p_{s}$ : coefficents
Permalink:: http://dlmf.nist.gov/13.24.E2
Encodings:: TeX, pMML, png

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 $\mathop{\exp\/}\nolimits\!\left(-\tfrac{1}{2}z\left(\mathop{\coth\/}\nolimits t% -\frac{1}{t}\right)\right)\left(\frac{t}{\mathop{\sinh\/}\nolimits t}\right)^{% {1-2\mu}}=\sum_{{s=0}}^{{\infty}}p_{s}^{{(\mu)}}(z)\left(-\frac{t}{z}\right)^{% s}.$

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, : nonnegative integer, : complex variable and $p_{s}$ : coefficents
Permalink:: http://dlmf.nist.gov/13.24.E3
Encodings:: TeX, pMML, png

(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.

§13.24 Series

Contents

§13.24(i) Expansions in Series of Whittaker Functions

§13.24(ii) Expansions in Series of Bessel Functions