28 GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 
and so by repetition, if m be integral, 
[1 K Pa Qo =— PQ Pin} 
l 
1 
n+1 
the analogue of Pii41Qn- QuisP, = 
Similarly from (4), putting «=1 and repeating we obtain 
2n-++5, r 2 1 
Prnpr(A) Qin (1) aw ui Qin+(A) Pinar) = as oe a : : (9) 
From the recurrence formulze we can obtain without difficulty 
r= Or +1 +1] Pray(A)Pim(A2) — Pina (A2)Pg(A) 
Sera Pe (A)P (0) oa a ; Hee Se ee D (¢) 
r=0 
A theorem analogous to NeuMaNN’s expansion of an arbitrary function in a series of 
Bessel Functions I have given in a paper, L..VW.S. Proceedings ; also the analogue of 
LomMEL’s theorem 
2 = {P+ HTP + STOP + «+ + ad ing, 
