# §30.4 Functions of the First Kind

## §30.4(i) Definitions

The eigenfunctions of (30.2.1) that correspond to the eigenvalues $\lambda^{m}_{n}\left(\gamma^{2}\right)$ are denoted by $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$, $n=m,m+1,m+2,\dots$. They are normalized by the condition

 30.4.1 $\int_{-1}^{1}\left(\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)\right)^{2}% \mathrm{d}x=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!},$

the sign of $\mathsf{Ps}^{m}_{n}\left(0,\gamma^{2}\right)$ being $(-1)^{(n+m)/2}$ when $n-m$ is even, and the sign of $\ifrac{\mathrm{d}\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)}{\mathrm{d}x}|_{% x=0}$ being $(-1)^{(n+m-1)/2}$ when $n-m$ is odd.

When $\gamma^{2}>0$ $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ is the prolate angular spheroidal wave function, and when $\gamma^{2}<0$ $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ is the oblate angular spheroidal wave function. If $\gamma=0$, $\mathsf{Ps}^{m}_{n}\left(x,0\right)$ reduces to the Ferrers function $\mathsf{P}^{m}_{n}\left(x\right)$:

 30.4.2 $\mathsf{Ps}^{m}_{n}\left(x,0\right)=\mathsf{P}^{m}_{n}\left(x\right);$

compare §14.3(i).

## §30.4(ii) Elementary Properties

 30.4.3 $\mathsf{Ps}^{m}_{n}\left(-x,\gamma^{2}\right)=(-1)^{n-m}\mathsf{Ps}^{m}_{n}% \left(x,\gamma^{2}\right).$

$\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ has exactly $n-m$ zeros in the interval $-1.

## §30.4(iii) Power-Series Expansion

 30.4.4 $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)=(1-x^{2})^{\frac{1}{2}m}\sum_{k=0% }^{\infty}g_{k}x^{k},$ $-1\leq x\leq 1$,

where

 30.4.5 $\alpha_{k}g_{k+2}+(\beta_{k}-\lambda^{m}_{n}\left(\gamma^{2}\right))g_{k}+% \gamma_{k}g_{k-2}=0$

with $\alpha_{k}$, $\beta_{k}$, $\gamma_{k}$ from (30.3.6), and $g_{-1}=g_{-2}=0$, $g_{k}=0$ for even $k$ if $n-m$ is odd and $g_{k}=0$ for odd $k$ if $n-m$ is even. Normalization of the coefficients $g_{k}$ is effected by application of (30.4.1).

## §30.4(iv) Orthogonality

 30.4.6 $\int_{-1}^{1}\mathsf{Ps}^{m}_{k}\left(x,\gamma^{2}\right)\mathsf{Ps}^{m}_{n}% \left(x,\gamma^{2}\right)\mathrm{d}x=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!}\delta% _{k,n}.$

If $f(x)$ is mean-square integrable on $[-1,1]$, then formally

 30.4.7 $f(x)=\sum_{n=m}^{\infty}c_{n}\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),$

where

 30.4.8 $c_{n}=(n+\tfrac{1}{2})\frac{(n-m)!}{(n+m)!}\int_{-1}^{1}f(t)\mathsf{Ps}^{m}_{n% }\left(t,\gamma^{2}\right)\mathrm{d}t.$

The expansion (30.4.7) converges in the norm of $L^{2}(-1,1)$, that is,

 30.4.9 $\lim_{N\to\infty}\int_{-1}^{1}{\left|f(x)-\sum_{n=m}^{N}c_{n}\mathsf{Ps}^{m}_{% n}\left(x,\gamma^{2}\right)\right|}^{2}\mathrm{d}x=0.$

It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for $-1\leq x\leq 1$.