182 
which will give, by (3), and by the principle last mentioned 
respecting odd exponents, 
(2X +1, 2, 2v) = {A; bs Me 
(2-1, 4p, 2») ={A—1) wy v)- 
We shall then have, by the mere notation, 
> pk =(r, py vp PPR” 5 (6) 
and, by treating this equation on the plan of (3), 
{A, ws v}=(A-1, py vj + (A, w—1,v}+ {Asp v1}. (7) 
By a precisely similar reasoning, attending only to j and &, or 
making \ = 0, we have an expression of the form, 
3 jel = (uy 9) Pa 8) 
where the coefficients {y, v} must satisfy the analogous equa- 
tion in differences, 
6) 
{u, vj={u—-1, v}+{uwv-1}, (9) 
together with the initial conditions, 
(a, 0}=1, (0, v)=1. (10) 
Hence, it is easy to infer that 
heal (ut+v)!. 
[Hs vj = leila (11) 
one way of obtaining which result is, to observe that the ge- 
nerating function has the form, 
S {u,v} u've’ = (1 -u—-v)". (12) 
In like manner, if we combine the equation in differences (7), 
with the initial conditions derived from the foregoing solution 
of a less complex problem, namely, with 
(0, sv} = (ps V}> {A, 0, vy} ={A, v}, {A, uw, O}={A,u}, (13) 
when the second members are interpreted as in (11), we find 
that the (slightly) more complex generating function sought is, 
= (A, wy, vjPute=(1-t-u-v)7; (14) 
and therefore that the required form of the coefficient is, 
a 
