[HARKNESS] THE ALGEBRAIC BASIS OF CERTAIN BESSEL SERIES 101 
It is useful to verify in a special case, say s=4, that if we start 
with the assumption that 1 can be expressed in the form 
pt1.p+2.p+3.p+4 pes (oem mme 2e nm: 
v+l.r+2.v+3.v+4 Z ee ETES 
ix p+3.. pt4 
2 Ta aera oO A 
p+4 : 
VeRO Voie TE D UE 
i 
AL pu rem em gsr, Gace 
where f, u, v, w are functions of » only, it is easy to find the value 
w=v+8 by putting p= —4; the value v=»+6 by putting p= —3, etc. 
As the biquadratic equation in p is satisfied by p=v as well as by 
p=—4, —3, —2, —1, the relation must be an identity. 
t 

1 
+ (p—v)s 
W, 
$ 3. The formula (6) may be written 
1 1 
Pega pce PPS a vel a2 apes 

yv+2 (p—v): 
si Le v+2...v+s+1l p+l 
v+4 (p— v)2 : 
te Ds En oa ee 
that is, 
1 + (p=—v): v)1 (p— v)2 
ee oO md El Spee 
HP Rae Rue 
LS 2 PES 
where the B’s depend on », but are independent of p. 
The general value of the coefficients B is given by 
ie y+2t 
Baie D: (ey ae, 
The B’s can be found by an independent method which gives in 
succession By, B,, Bs,.... For this purpose replace p successively 
by the values », y+1, v+2,.... This gives a set of linear equations. 
7 re +s 
1 
B ——————— ; 
ish 55 aa CE PETER NE 
Dial 1 
<8} ee rr SS ’ 
Tas a oT es Sa ay STE (7) 
2 ee 
Bit nr, 
y+4 y 
gee y+5 yv+4.7+5.7+6 
; 1 : 
Pate pH mener ct 
