2 THE REV. F. H. JACKSON ON 
reducing when p=1 to Gauss’s expression for I(n) or P(n + 1) 
; Me lee GES: 
+1) = nt1l-n+2-n+3 
The infinite product (4) is convergent for all values of n except negative integers : 
(2), will be in general 
TI, ([7]) 
Seah Wye E : 2 x 5 
ie ({7]) (p ) ( ) 
for the infinite product I,,([]) is 
Pak, Pe persk 
pl pal ee ee (a Len 
pent =| pm =a aig = 1 p =a 
oe jl) See Te 
which is 
ae ll ape eb ll ESE I. ya ow 0 pr+l1 Site || }) 0 aida 
iT prey a I ‘c ; : 
»([7]) x prt+ 1 pert l CF SM sk rte 4 ] pt 1 
II,({]) x Gviy 
(2), denoting a convergent infinite product reducing for integral values of n to 
(p+1)(p? +1)... (p"+1). 
The difference theorem for [I,([x]) is U,([#])=[a]II,([~—1]). The multiplication 
theorem which is the generalisation of 
m—-1 1 
Pain(w+ ‘) oes (e+ xz =) = T(nx) : (27) Ot oe 
Vb nN 
is investigated in another paper. 
As obtained from the differential equations the series are perfectly general with 
regard to n, and the question of [mn]! or II([m]) only arises in connection with the 
arbitrary constant multiplier of each series. . 
2. 
In a paper on generalised forms of the series of Bess—L and LEGENDRE (Proc. 
Edin. Math. Soc., vol. xxi.), 1 have shown that if J,,,(2) denote the series 
Ay gl) 4 ae ae 2 SS ea Bie nents } (6) 
[2] [2n+2] ° [2] [4] [2x +2] [2+ 4] ; ; 
then y =J,,,(z) satisfies a differential equation 
A 1!) fs .. ‘) 
poe gp tae Ell Ih — n)}o! +[n][— rly = ina”). : (7) 
dq) ; d d 
FPL denoting FE Ge) er 
