§30.5 Functions of the Second Kind

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

30.5.1		$${\mathsf{Qs}}_n^m(x,{\gamma }^2),$$
$n=m,m+1,m+2,\mathrm{\dots }$.
ⓘ Symbols: ${\mathsf{Qs}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: 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		$${\mathsf{Qs}}_n^m(-x,{\gamma }^2)={(-1)}^{n-m+1}{\mathsf{Qs}}_n^m(x,{\gamma }^2),$$
ⓘ Symbols: ${\mathsf{Qs}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: 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		$${\mathsf{Qs}}_n^m(x,0)={\mathsf{Q}}_n^m\left(x\right);$$
ⓘ Symbols: : Ferrers function of the second kind, ${\mathsf{Qs}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer and $n\ge m$: 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		$$𝒲\left\{{\mathsf{Ps}}_n^m(x,{\gamma }^2),{\mathsf{Qs}}_n^m(x,{\gamma }^2)\right\}=\frac{(n+m)!}{(1-x^2)(n-m)!}{A}_n^m({\gamma }^2)A_n^{-m}({\gamma }^2)(\ne 0),$$
ⓘ Symbols: $𝒲$: Wronskian, $!$: factorial (as in $n!$), ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, ${\mathsf{Qs}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: 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).