# §30.4 Functions of the First Kind

## §30.4(i) Definitions

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

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

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

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

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

compare §14.3(i).

## §30.4(ii) Elementary Properties

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

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

## §30.4(iii) Power-Series Expansion

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

where

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

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 $\int_{-1}^{1}\mathop{\mathsf{Ps}^{m}_{k}\/}\nolimits\!\left(x,\gamma^{2}\right% )\mathop{\mathsf{Ps}^{m}_{n}\/}\nolimits\!\left(x,\gamma^{2}\right)dx=\frac{2}% {2n+1}\frac{(n+m)!}{(n-m)!}\delta_{k,n}.$

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

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

where

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

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

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

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 $-1\leq x\leq 1$.