§30.5 Functions of the Second Kind

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

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

30.5.1 $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right),$ $n=m,m+1,m+2,\dots$ .

Symbols:: $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : 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

They satisfy

30.5.2 $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(-x,\gamma^{2}\right)=(-1)^{{n-m+1}}\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right),$

Symbols:: $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : 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

and

30.5.3 $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,0\right)=\mathop{\mathsf{Q}^{{m}}_{{n}}\/}\nolimits\!\left(x\right);$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : 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

compare §14.3(i). Also,

30.5.4 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right),\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)\right\}=\frac{(n+m)!}{(1-x^{2})(n-m)!}A_{n}^{m}(\gamma^{2})A_{n}^{{-m}}(\gamma^{2})\quad(\neq 0),$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $!$ : $n!$ : factorial, $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : 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

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

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