§30.8 Expansions in Series of Ferrers Functions

Keywords:: spheroidal wave functions
Referenced by:: §30.16(i)
Permalink:: http://dlmf.nist.gov/30.8

§30.8(i) Functions of the First Kind
§30.8(ii) Functions of the Second Kind

§30.8(i) Functions of the First Kind

Notes:: See Arscott (1964b, §8.2) and Meixner and Schäfke (1954, §§3.542, 3.62).
Keywords:: spheroidal wave functions
Referenced by:: §30.8(ii)
Permalink:: http://dlmf.nist.gov/30.8.i

30.8.1 $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)=\sum _{{k=-R}}^{{\infty}}(-1)^{k}a^{m}_{{n,k}}(\gamma^{2})\mathop{\mathsf{P}^{{m}}_{{n+2k}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $R=\left\lfloor(n-m)/2\right\rfloor$ : limit and $a^{m}_{{n,k}}(\gamma^{2})$ : coefficients
A&S Ref:: 21.7.1 (in different form)
Referenced by:: §30.16(ii), §30.16(ii)
Permalink:: http://dlmf.nist.gov/30.8.E1
Encodings:: TeX, pMML, png

where $\mathop{\mathsf{P}^{{m}}_{{n+2k}}\/}\nolimits\!\left(x\right)$ is the Ferrers function of the first kind (§14.3(i)), $R=\left\lfloor\frac{1}{2}(n-m)\right\rfloor$ , and the coefficients $a^{m}_{{n,k}}(\gamma^{2})$ are given by

30.8.2 $a^{m}_{{n,k}}(\gamma^{2})=(-1)^{k}\left(n+2k+\tfrac{1}{2}\right)\frac{(n-m+2k)!}{(n+m+2k)!}\*\int _{{-1}}^{1}\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)\mathop{\mathsf{P}^{{m}}_{{n+2k}}\/}\nolimits\!\left(x\right)dx.$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter and $a^{m}_{{n,k}}(\gamma^{2})$ : coefficients
Referenced by:: §30.11(i)
Permalink:: http://dlmf.nist.gov/30.8.E2
Encodings:: TeX, pMML, png

Let

30.8.3

$A_{k}=-\gamma^{2}\frac{(n-m+2k-1)(n-m+2k)}{(2n+4k-3)(2n+4k-1)},$

$B_{k}=(n+2k)(n+2k+1)-2\gamma^{2}\frac{(n+2k)(n+2k+1)-1+m^{2}}{(2n+4k-1)(2n+4k+3)},$

$C_{k}=-\gamma^{2}\frac{(n+m+2k+1)(n+m+2k+2)}{(2n+4k+3)(2n+4k+5)}.$

Symbols:: $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $A_{k}$ , $B_{k}$ and $C_{k}$
Permalink:: http://dlmf.nist.gov/30.8.E3
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

Then the set of coefficients $a^{m}_{{n,k}}(\gamma^{2})$ , $k=-R,-R+1,-R+2,\dots$ is the solution of the difference equation

30.8.4 $A_{k}f_{{k-1}}+\left(B_{k}-\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)\right)f_{k}+C_{k}f_{{k+1}}=0,$

Symbols:: $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $A_{k}$ , $B_{k}$ and $C_{k}$
Referenced by:: §30.16(ii), §30.8(ii)
Permalink:: http://dlmf.nist.gov/30.8.E4
Encodings:: TeX, pMML, png

(note that $A_{{-R}}=0$ ) that satisfies the normalizing condition

30.8.5 $\sum _{{k=-R}}^{{\infty}}a_{{n,k}}^{m}(\gamma^{2})a_{{n,k}}^{{-m}}(\gamma^{2})\frac{1}{2n+4k+1}=\frac{1}{2n+1},$

Symbols:: $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $R=\left\lfloor(n-m)/2\right\rfloor$ : limit and $a^{m}_{{n,k}}(\gamma^{2})$ : coefficients
Referenced by:: §30.16(ii)
Permalink:: http://dlmf.nist.gov/30.8.E5
Encodings:: TeX, pMML, png

with

30.8.6 $a_{{n,k}}^{{-m}}(\gamma^{2})=\frac{(n-m)!(n+m+2k)!}{(n+m)!(n-m+2k)!}a_{{n,k}}^{m}(\gamma^{2}).$

Symbols:: $!$ : $n!$ : factorial, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter and $a^{m}_{{n,k}}(\gamma^{2})$ : coefficients
Referenced by:: §30.11(i)
Permalink:: http://dlmf.nist.gov/30.8.E6
Encodings:: TeX, pMML, png

Also, as $k\to\infty$ ,

30.8.7 $\frac{k^{2}a^{m}_{{n,k}}(\gamma^{2})}{a^{m}_{{n,k-1}}(\gamma^{2})}=\frac{\gamma^{2}}{16}+\mathop{O\/}\nolimits\!\left(\frac{1}{k}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter and $a^{m}_{{n,k}}(\gamma^{2})$ : coefficients
Permalink:: http://dlmf.nist.gov/30.8.E7
Encodings:: TeX, pMML, png

and

30.8.8 $\frac{\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)-B_{k}}{A_{k}}\frac{a^{m}_{{n,k}}(\gamma^{2})}{a^{m}_{{n,k-1}}(\gamma^{2})}=1+\mathop{O\/}\nolimits\!\left(\frac{1}{k^{4}}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $a^{m}_{{n,k}}(\gamma^{2})$ : coefficients, $A_{k}$ and $B_{k}$
Permalink:: http://dlmf.nist.gov/30.8.E8
Encodings:: TeX, pMML, png

§30.8(ii) Functions of the Second Kind

Notes:: See Arscott (1964b, §8.5) and Meixner and Schäfke (1954, §3.62).
Keywords:: spheroidal wave functions
Referenced by:: §30.5
Permalink:: http://dlmf.nist.gov/30.8.ii

30.8.9 $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)=\sum _{{k=-\infty}}^{{-N-1}}(-1)^{k}{a^{{\prime}}}^{m}_{{n,k}}(\gamma^{2})\mathop{\mathsf{P}^{{m}}_{{n+2k}}\/}\nolimits\!\left(x\right)+\sum _{{k=-N}}^{{\infty}}(-1)^{k}a^{m}_{{n,k}}(\gamma^{2})\mathop{\mathsf{Q}^{{m}}_{{n+2k}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $a^{m}_{{n,k}}(\gamma^{2})$ : coefficients and $N$ : upper limit
A&S Ref:: 21.7.21 (in different form)
Permalink:: http://dlmf.nist.gov/30.8.E9
Encodings:: TeX, pMML, png

where $\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits$ and $\mathop{\mathsf{Q}^{{m}}_{{n}}\/}\nolimits$ are again the Ferrers functions and $N=\left\lfloor\frac{1}{2}(n+m)\right\rfloor$ . The coefficients $a^{m}_{{n,k}}(\gamma^{2})$ satisfy (30.8.4) for all $k$ when we set $a^{m}_{{n,k}}(\gamma^{2})=0$ for $k<-N$ . For $k\geq-R$ they agree with the coefficients defined in §30.8(i). For $k=-N,-N+1,\dots,-R-1$ they are determined from (30.8.4) by forward recursion using $a^{m}_{{n,-N-1}}(\gamma^{2})=0$ . The set of coefficients ${a^{{\prime}}}^{m}_{{n,k}}(\gamma^{2})$ , $k=-N-1,-N-2,\dots$ , is the recessive solution of (30.8.4) as $k\to-\infty$ that is normalized by

30.8.10 $A_{{-N-1}}{a^{{\prime}}}^{m}_{{n,-N-2}}(\gamma^{2})+{\left(B_{{-N-1}}-\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)\right){a^{{\prime}}}^{m}_{{n,-N-1}}(\gamma^{2})}+C^{{\prime}}a^{m}_{{n,-N}}(\gamma^{2})=0,$

Symbols:: $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter, $a^{m}_{{n,k}}(\gamma^{2})$ : coefficients, $A_{k}$ , $B_{k}$ , $N$ : upper limit and $C^{{\prime}}$
Permalink:: http://dlmf.nist.gov/30.8.E10
Encodings:: TeX, pMML, png

with

30.8.11 $C^{{\prime}}=\begin{cases}\dfrac{\gamma^{2}}{4m^{2}-1},&\mbox{$n-m$ even},\\ -\dfrac{\gamma^{2}}{(2m-1)(2m-3)},&\mbox{$n-m$ odd}.\end{cases}$

Symbols:: $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter and $C^{{\prime}}$
Permalink:: http://dlmf.nist.gov/30.8.E11
Encodings:: TeX, pMML, png

It should be noted that if the forward recursion (30.8.4) beginning with $f_{{-N-1}}=0$ , $f_{{-N}}=1$ leads to $f_{{-R}}=0$ , then $a^{m}_{{n,k}}(\gamma^{2})$ is undefined for $n<-R$ and $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ does not exist.

§30.8 Expansions in Series of Ferrers Functions

Contents

§30.8(i) Functions of the First Kind

§30.8(ii) Functions of the Second Kind