236
RESEARCHES ON THE PARTITION OF NUMBERS.
[140
where m' denotes any divisor whatever of m (unity and the number m itself not
excluded). Hence, if a represent a divisor of one or more of the indices m, and k
be the number of the indices of which a is a divisor, we have
fx = n [1 —
Now considering apart from the others one of the multiple factors [1 — # a ] fc , we
may write fx = [1 —
Suppose that the fraction is decomposed into simpler fractions, in the form
J x
%= I(X)
+ fa - + fa
+ &c.,
where I{x) denotes the integral part, and the &c. refers to the fractional terms
depending upon the other multiple factors such as [1 — x^f. The functions Ox are
to be considered as functions with indeterminate coefficients, the degree of each such
function being inferior by unity to that of the corresponding denominator; and it is
proper to remark that the number of the indeterminate coefficients in all the functions
Ox together is equal to the degree of the denominator fx.
The term {xd x ) k 1
Ox
[1 —x a ]
may be reduced to the form
gx g x x
[1 — x a ] k + [1 —
the functions gx being of the same degree as Ox, and the coefficients of these functions
being linearly connected with those of the function Ox. The first of the foregoing
terms is the only term on the right-hand side which contains the denominator [1 — off ;
hence, multiplying by this denominator and then writing [1 — ccP] = 0, we find
cbx
>- =gx,
fx
which is true when x is any root whatever of the equation [1 — x a ] = 0. Now by
means of the equation [1 — = 0, may be expressed in the form of a rational and
Ji x
integral function Gx, the degree of which is less by unity than that of [1—«®]. We
have therefore Gx = gx, an equation which is satisfied by each root of [1 — x a ~\ = 0,
and which is therefore an identical equation; gx is thus determined, and the coefficients
of 6x being linear functions of those of gx, the function Ox may be considered as
determined. And this being so, the function
(})X
f x
- f r àx) k 1
Ox
[1 — of]
