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30 Spheroidal Wave FunctionsProperties

§30.8 Expansions in Series of Ferrers Functions


§30.8(i) Functions of the First Kind

30.8.1 Psnm(x,γ2)=k=-R(-1)kan,km(γ2)Pn+2km(x),

where Pn+2km(x) is the Ferrers function of the first kind (§14.3(i)), R=12(n-m), and the coefficients an,km(γ2) are given by

30.8.2 an,km(γ2)=(-1)k(n+2k+12)(n-m+2k)!(n+m+2k)!-11Psnm(x,γ2)Pn+2km(x)dx.


30.8.3 Ak =-γ2(n-m+2k-1)(n-m+2k)(2n+4k-3)(2n+4k-1),
Bk =(n+2k)(n+2k+1)-2γ2(n+2k)(n+2k+1)-1+m2(2n+4k-1)(2n+4k+3),
Ck =-γ2(n+m+2k+1)(n+m+2k+2)(2n+4k+3)(2n+4k+5).

Then the set of coefficients an,km(γ2), k=-R,-R+1,-R+2, is the solution of the difference equation

30.8.4 Akfk-1+(Bk-λnm(γ2))fk+Ckfk+1=0,

(note that A-R=0) that satisfies the normalizing condition

30.8.5 k=-Ran,km(γ2)an,k-m(γ2)12n+4k+1=12n+1,


30.8.6 an,k-m(γ2)=(n-m)!(n+m+2k)!(n+m)!(n-m+2k)!an,km(γ2).

Also, as k,

30.8.7 k2an,km(γ2)an,k-1m(γ2)=γ216+O(1k),


30.8.8 λnm(γ2)-BkAkan,km(γ2)an,k-1m(γ2)=1+O(1k4).

§30.8(ii) Functions of the Second Kind

30.8.9 Qsnm(x,γ2)=k=--N-1(-1)kan,km(γ2)Pn+2km(x)+k=-N(-1)kan,km(γ2)Qn+2km(x),

where Pnm and Qnm are again the Ferrers functions and N=12(n+m). The coefficients an,km(γ2) satisfy (30.8.4) for all k when we set an,km(γ2)=0 for k<-N. For k-R they agree with the coefficients defined in §30.8(i). For k=-N,-N+1,,-R-1 they are determined from (30.8.4) by forward recursion using an,-N-1m(γ2)=0. The set of coefficients an,km(γ2), k=-N-1,-N-2,, is the recessive solution of (30.8.4) as k- that is normalized by

30.8.10 A-N-1an,-N-2m(γ2)+(B-N-1-λnm(γ2))an,-N-1m(γ2)+Can,-Nm(γ2)=0,


30.8.11 C={γ24m2-1,n-m even,-γ2(2m-1)(2m-3),n-m odd.

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 an,km(γ2) is undefined for n<-R and Qsnm(x,γ2) does not exist.