GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL, 27 
a Nive" 
= ; : : : 94 
ores (2)s(2)ntr ) 
Operating on this with D'™ we see that the operation reduces to zero all terms for 
which r<m, while the result of operating on the general term 
(2m+2r] 
[2] 
pial (cea) 
[m+r]! [2+ m+7]! (2)mirl2)nrmer (95) 
is “ 
[2m+2r] [2m+2r- 2] etn [2m + 2r — 2m + 2] Na Mo 
ane [2] [2] [2] (gett?) BI (96) 
[m+7r]! [n+ m+7]! (2)(2)n sme 
which is easily reduced to 
“SS p™ \ —[n+-m’ pases vO (n-+m-+2r] 
= Ge aint (8 : (97) 
[2]” [7]! [e+ m +7}! (2),(2)n4 mr 
viz., the general term of 
Nee mim+n ae ° 
pyr” ar Tales w) ; . (98) 
In a similar way theorem (90) may be generalised. 
We have shown that 
[2] - Poa(wA) =A[2n — 1]eP,, (eA) -— p" [nm - Pi, (@A) F (a) 
In the same way we can establish 
DY [m}Qi@A) = AL Ze — 1 JeQen aye", A) = [2 = TV] Qinn(@A) - (8) 
analogous to the ordinary recurrence formule. 
Multiplying (a) by Qin .(x?,A) and (8) by P,,-1(«”,A), then subtracting, we obtain 
[7] { Pin(%A) Qin—1(@?,r) bon fo a Ga eer Caer), } 
=[n- L]{Ppa(2?,A) Qn—n(X) =P" Qin—n(@,A)Pin—oy(@,A)} . (y) 
analogous to ' 
n{P,.Q,—1 = QPF} = (n I 1) {Pp 1Qn—2 re Qin iey=o) 
So also multiplying («) by Q,,-2(z,A) and (8) by p”*'P,,-2(#,A), then subtracting, we 
obtain 
[70] { Prn(%,A) Qin—n(@,A) = PQ in €X)Pin—(#A)} = AL Qn — 1 ]or{ Phi —1y(@?A) Qin—2(@X) — PQ in—1(4?,A) Pin—n(@A) } 
Seat Pin(@A) Qn #?,A) = P"*?Qn(@)A)Pin—(@"A) Fs . (8) 
When we put «=1 we obtain 
[7] { Pin(A) Qin) = PQ in A)Pin—n(A) } a [n a 1] { Pin—1(A) Qin—2(A) ey, P”* Qin—4(A) Pin—ay(A) } (€) 
