§30.5 Functions of the Second Kind

ⓘ

Defines:: $\mathsf{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the second kind
Keywords:: definitions, of the second kind, spheroidal wave functions
Notes:: See Meixner and Schäfke (1954, §3.6).
Permalink:: http://dlmf.nist.gov/30.5
See also:: Annotations for Ch.30

Other solutions of (30.2.1) with $\mu=m$ , $\lambda=\lambda^{m}_{n}\left(\gamma^{2}\right)$ , and $z=x$ are

30.5.1		$\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right),$
$n=m,m+1,m+2,\dots$ .
ⓘ Symbols: $\mathsf{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the second kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter Permalink: http://dlmf.nist.gov/30.5.E1 Encodings: TeX, pMML, png See also: Annotations for §30.5 and Ch.30

They satisfy

30.5.2		$\mathsf{Qs}^{m}_{n}\left(-x,\gamma^{2}\right)=(-1)^{n-m+1}\mathsf{Qs}^{m}_{n}% \left(x,\gamma^{2}\right),$
ⓘ Symbols: $\mathsf{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the second kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter Permalink: http://dlmf.nist.gov/30.5.E2 Encodings: TeX, pMML, png See also: Annotations for §30.5 and Ch.30

and

30.5.3		$\mathsf{Qs}^{m}_{n}\left(x,0\right)=\mathsf{Q}^{m}_{n}\left(x\right);$
ⓘ Symbols: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\mathsf{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the second kind, $x$ : real variable, $m$ : nonnegative integer and $n\geq m$ : integer degree Permalink: http://dlmf.nist.gov/30.5.E3 Encodings: TeX, pMML, png See also: Annotations for §30.5 and Ch.30

compare §14.3(i). Also,

30.5.4		$\mathscr{W}\left\{\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),\mathsf{Qs}^{m}% _{n}\left(x,\gamma^{2}\right)\right\}=\frac{(n+m)!}{(1-x^{2})(n-m)!}A_{n}^{m}(% \gamma^{2})A_{n}^{-m}(\gamma^{2})\quad(\neq 0),$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $!$ : factorial (as in $n!$ ), $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, $\mathsf{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the second kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $A_{n}^{m}(\gamma^{2})$ and $\gamma^{2}$ : real parameter Permalink: http://dlmf.nist.gov/30.5.E4 Encodings: TeX, pMML, png See also: Annotations for §30.5 and Ch.30

with $A_{n}^{\pm m}(\gamma^{2})$ as in (30.11.4).

For further properties see Meixner and Schäfke (1954) and §30.8(ii).