[HARKNESS] THE ALGEBRAIC BASIS OF CERTAIN BESSEL SERIES 99 
ve p+l.p+2...p+s (5) ex p+2...p+s ie 
py+l.. 
v.v+l.v+2...v+s ..v+s+1 
p+3 eae 
Ae JG np UE 
NN em (een (6) 
Te ee ere gy IMC 
where we use () as the equivalent of C,. The value of @) will 
be taken to be 1. 
The identity (5) may be proved as fol'ows: 
Assume that it holds for s—1, so ian 
re eer / (ov) à y+ 1) 
p+1.p+2...p+s—1 Ey L ani @ SS a ee 
aed yt1.v+2..v+s—1 De OF Ve 
Piaeee Doro Dunes p+2 
s—1 p—v+2). 
eg ) oertes p+3 0” 
Then 
v+1.yv+2...v+s 
p+1.p+2...p+s 
oy y+1.yv4+2...»v+s—1., GC) 

p+1.p+2...p+s—1 pts 
v+l.v+2...v+s—1l Sg say ead int GON Ane eal aoe 
pea ee Pale Ghat nPop los 
(os See aa D 
Gee 1 p+1 Ge pt1. p+2 
TUE Por) Fa 
en EE mm 
> ca (o=»)1 se Ge HG an end DE 
o J} pri p+1.p+2 
CNRS (PE var) sen se 
LAINE AE ve ES) 
= =1=G = Un aes a 
Se (SS Re) a 
pot pa 2: p43 
Thus if (5) holds for s—1 it must also hold for s. As the identity is 
true for s=1, 2, 3, it follows by mathematical induction that it is true 
generally. 
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