108 THE ROYAL SOCIETY: OF CANADA 
§ 8. As an illustration of the use to which formula (17) can be put, 
we shall examine whether there is any formula for J2,(2x) analogous 
to the formula 
ee) 
2 J (x) = 2 CRT OA OS (real part of »>0) 
a formula which Nielsen singles out as “remarkable”” (N. p. 299). 
In Schénholzer’s formula for the product of two J’s, see (2) § 1, we 
write v+n and »y—n for p. Hence 
SJ y40 (x) er (x) = 
x \ 22s 
= CDS GES yv+s.v+ts—1...v+s—n+l (5) 
=, s p+s+ 1. v+s+2...v+s+n T(v+s+1)? 
It follows that 
2 PAGE) EE GT NO) EE AE) ane) 
ne Je (R)-J> ei] 
Hyde re 2v+2s 1 vts.v+ts—1 
epi 1 ( s een [i yts+1 .v-+s+2 
49 vs. pv+s—1l.v+s—2.y+s—3 et & 21725 
v+stl .vts42.v+543.7+54+4 5° °° al iS) 
pee ee ores 1 Tov+s)Po+st+1) x2+2s 
=2E ( 1 ( : Yen te 
: T'(2v-F2s) 4 
=2 > oe ep nr ho a SLE Prat er 1 LS 
so T(2+s+D (stl) vts T(2»+25) (5) 
D x2r +25 
Re mere en eee Ts) 
In the special case v=0, Jo(2x) must be replaced by 1+Jo(2x). This 
can be verified at once by combining 
Jo(2x) = Jo?(x) — 27, (x) + 2So?(x) —27$(x) +... 
1= J5(x) +27,2(x) +2752(x) +273 (x) +... 
(G. and M. pp. 28, 29). 
The formula at which we have thus arrived, namely 
pl Sree) 
with 
ee] 
Jr) 2 AGE MENT ES) Te) (19) 
É 1 
S= 
is clearly constructed after the same pattern as that quoted above for 
à J?, (a) 
2 y : 
In the special case where » is a positive integer m, an independent 
proof can be based on the equation 
Se 
22 | i) = Jy (x) +Si (x) t+Jo(x)O +... +I (x) +... 
1 
FAO) tI Oa 
