154] SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS. 509
blit this equation gives
and we have
[1 — x a ] = (x — p) (x — p a *) ... (x — p a a),
if 1, a 2 , ••• a a are the integers less than a and prime to it (a is of course the degree
of [1 — ¿e a ]). Hence
XP = 6p p a ~' n (1 - p«*- 1 ) ’
and therefore
0p = - n (1 - p^- 1 ) xp ;
or putting for XP its value
dp = — p a n (1 — p ai_1 ) A_ s ,
where a is the degree of [1 — x a ~\ and
of unity) less than a and prime to it.
the equation [1 — p a ] = 0, may be reduced to
a—1, and then by simply changing p into x
Occ
then by multiplication
cii denotes in succession the integers (exclusive
The function on the right hand, by means of
an integral function of p of the degree
we have the required function 6x. The
fraction
can
of the terms by the proper factor be
[1 — # a ]
reduced to a fraction with the denominator 1 — x a , and the coefficients of the numerator
of this fraction are the coefficients of the corresponding prime circulator ( ) per a q .
Thus, let it be required to find the terms depending on the denominator [1 — af\ in
1
these are
where
(1 - x) (1 - x 2 ) (1 - x 3 ) (1 -x i )(l- x 5 ) (1 - x e ) ’
S-X&-, xd x S
p — x p — X
YjP = coeff. \ in v, P -,
t f(per t ) >
X?p = coeff. ^ in t P
t f(p e ~ l )
1 1
f(pe- f ) " (1 - per*) (1 - p 2 e- 2i ) (1 - p 4 e~ 4i ) (1 - p 5 e~ 5t ) (1 - e~ st ) (1 - e~ 6t )
and