GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 15 
When m is integral and positive [P*z]” reduces to the product of m factors 
V@zeie ee)... ... (Pei 4.1) 
Dividing both sides of (10) by [P*z]”" we have 
[Petz] ee DY Sys oa ne nisl Pea Rye 3 Ss: Reet 
Tae 7 PEADGA ot ots Po] Py, ., Peery MY) 
putting er 
He n—-vV 
2 
al 
y = —-n+v+1 
x=nt+v+l 
the general term of the series becomes 
(—aypritenP’ a1 Ene ea aes Pate tel 
P? oe 1 Wh ae Pp as 1 il = p2r-1 Re Sear: jl a pert 
(allyines [m—v][m—v-l1]...... [n-v-2r+1] _, 
CAS. wien oeer) 
which is the r+1™ term in, Const. x P’ (1. p*~’). 
U+Y » |mN e 
The infinite product ae when we make the same substitutions as in the series, 
becomes 
(prt — leer = 1) oa: (Chis i 1) : (Bee ye 1) Perel) (Rea ae 1) (12) 
is (R® = ee = 1) UT (eyeee wy 1) ; (Rea = 1) cpus (pes = 1) 
Pssil 
which since 
[PSs ARES We ee oe [peri =, || 
re tees aye catia) (PPD) os. . (Pak +1) 
and 
p-1 Sale pr” =F || se Ppee—2K+1 _ 1 
i (dee _ (ees ms, 1) ae pe! (eae aa 1) 
(PP — NYO cae = 1) ati ee (Bzeaect2 ~ 1) 
we are justified in writing 
IE({m +v])I({x}) - (2)n ; ; f We. (13) 
T({2n]})1((v])(2), 
by analogy with Gauss’s II function, 
Moreover, when n and » are positive and integral, the infinite product reduces to 
[n-+r]}! [nr]! (2), 
“Pj DT), 
We have now 
T((m+vJ)M((r}(Qn pi? =i ie Finca BE : ~ (14) 
T1([2n])M([v])(2), (2) 22-1) 
