# §30.8 Expansions in Series of Ferrers Functions

## §30.8(i) Functions of the First Kind

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

where $\mathsf{P}^{m}_{n+2k}\left(x\right)$ 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 $a^{m}_{n,k}(\gamma^{2})=(-1)^{k}\left(n+2k+\tfrac{1}{2}\right)\frac{(n-m+2k)!}% {(n+m+2k)!}\*\int_{-1}^{1}\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)\mathsf{% P}^{m}_{n+2k}\left(x\right)\mathrm{d}x.$ ⓘ Defines: $a^{m}_{n,k}(\gamma^{2})$: coefficients (locally) Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the first kind, $\mathrm{d}\NVar{x}$: differential, $!$: factorial (as in $n!$), $\int$: integral, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\geq m$: integer degree and $\gamma^{2}$: real parameter Referenced by: §30.11(i) Permalink: http://dlmf.nist.gov/30.8.E2 Encodings: TeX, pMML, png See also: Annotations for §30.8(i), §30.8 and Ch.30

Let

 30.8.3 $\displaystyle A_{k}$ $\displaystyle=-\gamma^{2}\frac{(n-m+2k-1)(n-m+2k)}{(2n+4k-3)(2n+4k-1)},$ $\displaystyle B_{k}$ $\displaystyle=(n+2k)(n+2k+1)-2\gamma^{2}\frac{(n+2k)(n+2k+1)-1+m^{2}}{(2n+4k-1% )(2n+4k+3)},$ $\displaystyle C_{k}$ $\displaystyle=-\gamma^{2}\frac{(n+m+2k+1)(n+m+2k+2)}{(2n+4k+3)(2n+4k+5)}.$ ⓘ Defines: $A_{k}$ (locally), $B_{k}$ (locally) and $C_{k}$ (locally) Symbols: $m$: nonnegative integer, $n\geq m$: integer degree and $\gamma^{2}$: real parameter Permalink: http://dlmf.nist.gov/30.8.E3 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §30.8(i), §30.8 and Ch.30

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

(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 See also: Annotations for §30.8(i), §30.8 and Ch.30

with

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

Also, as $k\to\infty$,

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

and

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

## §30.8(ii) Functions of the Second Kind

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

where $\mathsf{P}^{m}_{n}$ and $\mathsf{Q}^{m}_{n}$ 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 $A_{-N-1}{a^{\prime}}^{m}_{n,-N-2}(\gamma^{2})+{\left(B_{-N-1}-\lambda^{m}_{n}% \left(\gamma^{2}\right)\right){a^{\prime}}^{m}_{n,-N-1}(\gamma^{2})}+C^{\prime% }a^{m}_{n,-N}(\gamma^{2})=0,$

with

 30.8.11 $C^{\prime}=\begin{cases}\dfrac{\gamma^{2}}{4m^{2}-1},&\mbox{n-m even},\\ -\dfrac{\gamma^{2}}{(2m-1)(2m-3)},&\mbox{n-m odd}.\end{cases}$ ⓘ Defines: $C^{\prime}$ (locally) Symbols: $m$: nonnegative integer, $n\geq m$: integer degree and $\gamma^{2}$: real parameter Permalink: http://dlmf.nist.gov/30.8.E11 Encodings: TeX, pMML, png See also: Annotations for §30.8(ii), §30.8 and Ch.30

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 $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)$ does not exist.