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§30.8 Expansions in Series of Ferrers Functions

  1. §30.8(i) Functions of the First Kind
  2. §30.8(ii) Functions of the Second Kind

§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(nm), and the coefficients an,km(γ2) are given by

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


30.8.3 Ak =γ2(nm+2k1)(nm+2k)(2n+4k3)(2n+4k1),
Bk =(n+2k)(n+2k+1)2γ2(n+2k)(n+2k+1)1+m2(2n+4k1)(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 Akfk1+(Bkλnm(γ2))fk+Ckfk+1=0,

(note that AR=0) that satisfies the normalizing condition

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


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

Also, as k,

30.8.7 k2an,km(γ2)an,k1m(γ2)=γ216+O(1k),


30.8.8 λnm(γ2)BkAkan,km(γ2)an,k1m(γ2)=1+O(1k4).

§30.8(ii) Functions of the Second Kind

30.8.9 𝖰𝗌nm(x,γ2)=k=N1(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 kR they agree with the coefficients defined in §30.8(i). For k=N,N+1,,R1 they are determined from (30.8.4) by forward recursion using an,N1m(γ2)=0. The set of coefficients an,km(γ2), k=N1,N2,, is the recessive solution of (30.8.4) as k that is normalized by

30.8.10 AN1an,N2m(γ2)+(BN1λnm(γ2))an,N1m(γ2)+Can,Nm(γ2)=0,


30.8.11 C={γ24m21,nm even,γ2(2m1)(2m3),nm odd.

It should be noted that if the forward recursion (30.8.4) beginning with fN1=0, fN=1 leads to fR=0, then an,km(γ2) is undefined for n<R and 𝖰𝗌nm(x,γ2) does not exist.