GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 21 
which is the general term of 
nz=v.n-v—2 [n-v][n-v—-2].... [2 
Ap. 4 n—v—1][n-v-3] sae [ nk Per) C . (48) 
If we had used the value m, =1 
Mr = 27 +1 
we should have obtained a series satisfying the differential equation (39). 
The series being 
y= Af a = Poe Palen \ . (49) 
the general term being 
oe \\e—v— art 7 (w—v— 1) eee 
(2 pre = 7A Ale Amat ita | [7 Es 2] [may 2) onan (50) 
If n—v be an odd integer and the series is finite, the number of terms will be 
m-v+l 
2 
: n-v—1 
For + substitute —{—-r 
The general term then becomes 
A [w-v—-l1][n-v-3].... [2] n—»—2r re [2n — 27°]! _ qf-'-2 (51) 
n—v+3.n—v+1 
Ap 4 [m+v][mt+v-2]..... [1 | [n—r]! [n—v —2r}! [7]! (2)n—(2), 
showing that the series is 
const, x P/, (aA) : : : 2 (52) 
6. 
In this and the following articles we shall give examples of the expansion of various 
functions in infinite series of the generalised Bessel’s Functions. The three expansions 
to be considered are analogous to the following theorems in ordinary Bessel’s Functions, 
= n+l »N+2 
a“ aw wv 
J (x) = ar, id ol®) + FAs yp 11") ar Pp ON (x) at Oe do to Pgh os 5 (53) 
m num +1 
S"J,(2) = 2" { Tat) + FT nga(t) + gp Imaal) too see (54) 
Ae x x 
amt Un) + aI ntalt) topo ne)t «+ : = 5) 
the symbol 8” denoting m successive integrations in which no arbitrary constants 
are introduced (TopHuNTER’s Functions of Laplace, Bessel, and Legendre, art’s. 
418-422), 
When [2] denotes — oe! prt) 
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