THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION. 107 
By means of (5) we write this— 
ee ee (Le 
Vleet (ery Harb} aryl 
EAN 42 {47}! eee. he 7 
‘eee ake rae + Torey} ar} ar 2} } (7) 
rr 47'] ee Nr 4 ( 
: oH pene ee 
We see by inspection that the coefficient of \” vanishes: the coetticient of \” is the 
expression 
Gi {4r}! ie [4] (2) pol8] [27] [2r-2] 
{Qr} {oF 1{ Ir} {Ir} [2] [ar+2]~ Pig] [27 +2] [2r+4] _ 
Bote seen (Gall fecal ileal ee ee « (4) [2) ; 
ere be [Qr+2)[Qr+4]........ el ae 
The series within the large brackets is easily summed term by term. The sum of the 
first two terms is =e = which is a factor of the third term. The sum of 
[27 - 2] , 
49) [20+ 2] and the third term is 
oL27 — 4] [27 - 2] 
[2r +2] [2r+4] 
Continuing in this way, we obtain that the sum of the first 7 terms is 
(= yp [2r = 2] (2r= 4) [2r— 6]... (4) [2] 
P2areerai) Dare |r SG ee. sn ola [4r — 2] 
which is equal to the last term, but is of opposite sign. The coefficient of X” is zero, 
and only the constant unity is left. Therefore 
L = Sp(A)apoi(A) be mA)dn(A)+ 2. ee ee + ET (ABA “ace ia (9) 
which, if p=1, reduces to 
= gp a Ou eeepc is « et even : ; 5 (Gl) 
4. 
Consider now the series 
Tid) di) — LE sys) Sp oorepseas, tepect cake pial) ef hele eae = paneled) 
Referring to expression (8), we see that the coetticient of A”” is 
ers }! 4 27] 8 20 2] 3 
(ore rere (2) Geral" a] Beayareyt ies 
