140] 
RESEARCHES ON THE PARTITION OF NUMBERS. 
245 
(S+l) 
which leads to 
P'(0, 1, 2 ... k) m ^ (km —a) = 
S s i(—V coeff. x& k ~ s)m in t 
( (1 - X 2 ) ... (1 -tf s )(l -#)(1 -X 2 ) ... (1 
where the summation extends, as in the former case, from s = 0 to the greatest value 
of s, for which (£& — s)m — |a-^s(s + 1) is positive or zero, or, if we please, when k is 
even, from 5 = 0 to s = \k—l, and when s is odd, from 5 = 0 to s = \ (k -1). The 
condition, in order that the fraction may be a proper one for all values of s, is. 
when k is even, a+l <\k(k+ 2), and when k is odd, a + 1 < \(k + 1) (k + 3). 
To transform the preceding expressions, I write when k is odd x 2 instead of x, 
and I put for shortness 6 instead of %k-s or 2(\k-s), and y instead of §a + ^s(5 + l) 
or a + 5(s + l); we have to consider an expression of the form 
coefficient x 6m in 
xy 
Tx' 
where Fx is the product of factors of the form 1 - x a . Suppose that a is the least 
common multiple of a and 6, then (1 - x a ') = (1 - x a ) is an integral function of x, 
equal x x suppose, and 1 -f- (1 — x a ) = x x + (1 - & a ')- Making this change in all the 
factors of Fx which require it (i.e. in all the factors except those in which a is a 
multiple of 6), the general term becomes 
coefficient x 6m in 
x y Hx 
~Gx 
where Gx is a product of factors of the form 1 — x M ', in which a' is a multiple of 0, 
i.e. Gx is a rational and integral function of x 9 . But in the numerator x y lix we may 
reject, as not contributing to the formation of the coefficient of x 9m , all the terms in 
which the indices are not multiples of 0; the numerator is thus reduced to a rational 
and integral function of x e , and the general term is therefore of the form 
coefficient x 6m in 
\(x e ) 
k^Y 
or what is the same thing, of the form 
coefficient x m in 
\x 
KX 
where kx is the product of factors of the form 1 — x a , and Xx is a rational and integral 
function of x. The particular value of the fraction depends on the value of 5; and 
uniting the different terms, vve have an expression 
coefficient x m in S s (—) s 
which is equivalent to 
coefficient x m in 
d>x 
fx ’