246
RESEARCHES ON THE PARTITION OF NUMBERS.
[140
where fx is a product of factors of the form 1 — x a , and cf>x is a rational and integral
function of x. And it is clear that the fraction will be a proper one when each
of the fractions in the original expression is a proper fraction, i.e. in the case of
P(0, 1, 2 ... k) m \(km — cl), when for k even, a<\k(k+2.), and for k odd, a<i(& + l)(/c+3);
and in the case of P'(0> 1, 2 ... k) m \{km-cl), when for k even, a+1 < %k (k+ 2), and
for k odd, cl + 1 < | (k + 1) (k + 3).
We see, therefore, that
and
are each of them of the form
P (0, 1, 2 ... k) m \{km — a),
P' (0, 1, 2 ... k) m km — cl),
coefficient x m in
(px
f*'
where fx is the product of factors of the form 1 — x a , and up to certain limiting values
of a the fraction is a proper fraction. When the fraction ^ is known, we may there-
jx
fore obtain by the method employed in the former part of this Memoir, analytical
expressions (involving prime circulators) for the functions P and P'.
As an example, take
which is equal to
coefficient x 3m in
— coefficient x m in
P(0, 1, 2, 3) m \m,
1
(1 - X 2 ) (1 - X i ) (1 - X 6 )
1
(1 - X 2 ) (1 - X 2 ) (1 -~P)'
The multiplier for the first fraction is
which is equal to
(1 - x 6 ) (1 - a; 12 )
(1 — X 2 ) (1 — £C 4 ) ’
1 + x 2 + 2x * i + ocf + 2 ¿c 8 + x 10 + x 12 .
Hence, rejecting in the numerator the terms the indices of which are not divisible
by 3, the first term becomes
coefficient x 3m in
1 + of + x 12
(1 - a?) (1 - x 12 ) (1 - x 6 ) ’
or what is the same thing, the first term is
coefficient x m in
1 + X 2 + X*
(l—a 2 ) 2 (i—O’