§30.8 Expansions in Series of Ferrers Functions

30.8.1		$${\mathsf{Ps}}_n^m(x,{\gamma }^2)=\sum _{k=-R}^{\mathrm{\infty }}{(-1)}^k{a}_{n,k}^m({\gamma }^2){\mathsf{P}}_{n+2k}^m\left(x\right),$$
ⓘ Symbols: : Ferrers function of the first kind, ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $R=\lfloor (n-m)/2\rfloor$: limit and $a_{n,k}^m({\gamma }^2)$: 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 See also: Annotations for §30.8(i), §30.8 and Ch.30

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

30.8.2		$$a_{n,k}^m({\gamma }^2)={(-1)}^k\left(n+2k+\frac{1}{2}\right)\frac{(n-m+2k)!}{(n+m+2k)!}{\int }_{-1}^1{\mathsf{Ps}}_n^m(x,{\gamma }^2){\mathsf{P}}_{n+2k}^m\left(x\right)dx.$$
ⓘ Defines: $a_{n,k}^m({\gamma }^2)$: coefficients (locally) Symbols: : Ferrers function of the first kind, $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: 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		$A_k$	$=-{\gamma }^2{\displaystyle \frac{(n-m+2k-1)(n-m+2k)}{(2n+4k-3)(2n+4k-1)}},$
		$B_k$	$=(n+2k)(n+2k+1)-2{\gamma }^2{\displaystyle \frac{(n+2k)(n+2k+1)-1+m^2}{(2n+4k-1)(2n+4k+3)}},$
		$C_k$	$=-{\gamma }^2{\displaystyle \frac{(n+m+2k+1)(n+m+2k+2)}{(2n+4k+3)(2n+4k+5)}}.$
ⓘ Defines: $A_k$ (locally), $B_k$ (locally) and $C_k$ (locally) Symbols: $m$: nonnegative integer, $n\ge m$: 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 $a_{n,k}^m({\gamma }^2)$, $k=-R,-R+1,-R+2,\mathrm{\dots }$ is the solution of the difference equation

30.8.4		$$A_k{f}_{k-1}+\left(B_k-{\lambda }_n^m\left({\gamma }^2\right)\right)f_k+C_k{f}_{k+1}=0,$$
ⓘ Symbols: ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: 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 See also: Annotations for §30.8(i), §30.8 and Ch.30

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

30.8.5		$$\sum _{k=-R}^{\mathrm{\infty }}{a}_{n,k}^m({\gamma }^2)a_{n,k}^{-m}({\gamma }^2)\frac{1}{2n+4k+1}=\frac{1}{2n+1},$$
ⓘ Symbols: $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $R=\lfloor (n-m)/2\rfloor$: limit and $a_{n,k}^m({\gamma }^2)$: 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		$$a_{n,k}^{-m}({\gamma }^2)=\frac{(n-m)!(n+m+2k)!}{(n+m)!(n-m+2k)!}{a}_{n,k}^m({\gamma }^2).$$
ⓘ Symbols: $!$: factorial (as in $n!$), $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients Referenced by: §30.11(i) Permalink: http://dlmf.nist.gov/30.8.E6 Encodings: TeX, pMML, png See also: Annotations for §30.8(i), §30.8 and Ch.30

Also, as $k\to \mathrm{\infty }$,

30.8.7		$$\frac{k^2{a}_{n,k}^m({\gamma }^2)}{a_{n,k-1}^m({\gamma }^2)}=\frac{{\gamma }^2}{16}+O\left(\frac{1}{k}\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients Permalink: http://dlmf.nist.gov/30.8.E7 Encodings: TeX, pMML, png See also: Annotations for §30.8(i), §30.8 and Ch.30

and

30.8.8		$$\frac{{\lambda }_n^m\left({\gamma }^2\right)-B_k}{A_k}\frac{a_{n,k}^m({\gamma }^2)}{a_{n,k-1}^m({\gamma }^2)}=1+O\left(\frac{1}{k^4}\right).$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $a_{n,k}^m({\gamma }^2)$: coefficients, $A_k$ and $B_k$ Permalink: http://dlmf.nist.gov/30.8.E8 Encodings: TeX, pMML, png See also: Annotations for §30.8(i), §30.8 and Ch.30

30.8.9		$${\mathsf{Qs}}_n^m(x,{\gamma }^2)=\sum _{k=-\mathrm{\infty }}^{-N-1}{(-1)}^{k}a^{\prime }{}_{n,k}{}^m({\gamma }^2){\mathsf{P}}_{n+2k}^m\left(x\right)+\sum _{k=-N}^{\mathrm{\infty }}{(-1)}^k{a}_{n,k}^m({\gamma }^2){\mathsf{Q}}_{n+2k}^m\left(x\right),$$
ⓘ Symbols: : Ferrers function of the first kind, : 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, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $a_{n,k}^m({\gamma }^2)$: 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 See also: Annotations for §30.8(ii), §30.8 and Ch.30

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

30.8.10		$$A_{-N-1}a^{\prime }{}_{n,-N-2}{}^m({\gamma }^2)+\left(B_{-N-1}-{\lambda }_n^m\left({\gamma }^2\right)\right)a^{\prime }{}_{n,-N-1}{}^m({\gamma }^2)+C^{\prime }{a}_{n,-N}^m({\gamma }^2)=0,$$
ⓘ Symbols: ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $a_{n,k}^m({\gamma }^2)$: coefficients, $A_k$, $B_k$, $N$: upper limit and $C^{\prime }$ Permalink: http://dlmf.nist.gov/30.8.E10 Encodings: TeX, pMML, png See also: Annotations for §30.8(ii), §30.8 and Ch.30

with

30.8.11
ⓘ Defines: $C^{\prime }$ (locally) Symbols: $m$: nonnegative integer, $n\ge m$: 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 $a_{n,k}^m({\gamma }^2)$ is undefined for and ${\mathsf{Qs}}_n^m(x,{\gamma }^2)$ does not exist.