§30.8 Expansions in Series of Ferrers Functions

where ${𝖯}_{n+2k}^m\left(x\right)$ 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

Let

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

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

with

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

and

where ${𝖯}_n^m$ and ${𝖰}_n^m$ 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

with

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 ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$ does not exist.