Greinar (Vísindafélag Íslendinga) - 01.01.1935, Síða 100

98
[6]
Now, in (lla) we can put n== — q, then we have
y(-q) = x'i-p z(-P) (14)
Multiplying (10) into xq_P and assuming n= — p we ob-
tain, having due regard to (14):
xq-P y(-P) (p—q)! xq+1_p y(1_p)
/(-q)=
(q—p)! (p—q-l)!(q—P+1)!
(p—q)! xq+2_p y(2-P) (p—q)! xq+3-p y(3-p)
(15)
1 2! (p q 2)! (q-p+2)! 1 3! (p - q—3)! (q-p+3)+'
This formula is an identity, valid for any function which
can be expanded into a power series of x where the
first term of the series y(-q) has xq-p as a factor; this may
be concluded from (14). However, we may easily render
(15) quite general by adding the first q—p terms from
the Maclaurin’s Series. The general formula is therefore:
y(_q)=yo(_q) +x y0(1_q)+ 2! y0(2_q) + • •
x^p-1 y0(~p_1)
(q—p-l)!
Xq P y(~P)
+ ,
(p—q)! xq+1 p y
(i-P)
(q—p)! 1 (p-q—l)!(q—p+1)!
(p—q)! xq+2-p y(2_p)
2!(p-q-2)!(q-p+2)Í + "' (15a)
From (15a) we get various series, according to the
values given to p and q. For instance, with q=1, and
p=0, —1, —2 etc., we have the following series:
y dx:
j y dx= xy
i x2 dy
x2 dy , x3 d2 y
2! dx 3! dx2
2 x3 d2 y , 3x4 d3 y
3! dx2
y dx=xy0+
and so on.
„)+I.2.x*d>y
4! dx* 1
2 ■ 3 x4 d3 y
2-3! dx2 2-4! dx3
+ ■
(16a)
(I6b)
(16c)
The first of these series, (16a), is the usual series ob-
tained by partial integration, and is therefore well known.
The other series seem to be Iess known, but they are all