GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 23 
From which we can see at once that 
[7] 5 AMtigint n+1 
en) = OnE per RM) Death [m "en Q),y Jin a” PN) 
a ins ae on Pi) tes (60) 
the general term being 
— 7 )S8p%-s+1 Nia aes [x] [w+ 1] SERS [w+s— 1] (2)nte—1 i res 
a @),der. [I Oa. 
subject to the convergence of the series; for if out of each term of the series on the 
right side of (60) we pick out the part involving \"*”a"**", we get a set of r+ 1 terms 
n+2r[n-+2r 1 1 = 2 1 n (2), 1 
a TO. AOS. ~ “wea ota FPO, 
ne ane ; rae . en) 
which by (59) reduces to 
LePrgylr+2)- (62) 
[a+r]! [7]! (2)n4r(2), 
the general term of J,,,(#A),—and theorem (60) is established. 
A particular case of this is 
3) 
R2q11 ” a 
Jm(wd) = prey o(t?”A) — p? BCE a) Play = oy, 10 ae ae . (63) 
analogous to 
ape Jy 
J,(a) = = Toa) + SG H@ +--+. 00+ . (64) 
To investigate a theorem analogous to 
S”.J,(x) = an T(t) + Ty uo(ae) + — Olan As oe i 2) (GD) 
If, in the series on the right, we replace the Bessel’s Functions by infinite series and 
collect the terms together according to powers of x, we find that the terms involving 
«z”* form an infinite series 
gingntir $ 1 m il mm+1 1 : \ (66) 
merirl arr im+trt ll r— 1! oe 2! ee ie ee 
which is 
DMgm+2r ° ‘ 
a : aie ee eat Ae LY 6D 
2H! |) mere mter—l-r+1 Meretaritliey AS oe r+1 J 
The series within the bracket is by an extension of the notation of VANDERMONDE’S 
theorem (Proc. Lond. Math. Soc., vol. xxvi. p. 285), 
Cam = Fe) mal, + = ad (eee . (68) 
which is (r+7)_, subject to convergence conditions, viz., 2 +1>0 
1 
OOS SCI 5 ry ar Ts a 
