Rit (Vísindafélag Íslendinga) - 01.06.1946, Blaðsíða 15

15
be left out of consideration. Some may find the aspects of
the equations expressed by (17) and (18) rather abhorrent,
it is therefore fortunate that the matter may be approached
from a slightly different point of view. I shall then write
down the general formula for asymptotic differentiations:

t d*y
dx
iH — p)\ xP (fy_
(k —1)\ dxp
y- / (ft - p)\ _ (n — p)\ \ f ffJo
^öl (k — t)! (n — t)\!{n—p)\ dxn
(19)
which is readily obtained from the Maclaurin series for
d‘y dpp dky0
—r and —- by elimination of . In particular we
dx dx dx
have
^ fy k\ ^-1 k\ n\ \ xn dny0
dx* (k — t) 1 ^ „ = o \(k — t)\ (n — t)\l n\ dx'1
(20)
If we in the infinite series on the right hand sides of (19)
and (20) let the remainder include the terms after n — k
an expression for this remainder is easily obtainable by
means of (7). In connexion with (19) and (20) the fol-
lowing transformation frequentiy is useful:
(k — p)\ _ (n—p) 1
(k — t)\ (n —1)\
_ y-________(n — p)! (t — p)\ (k — n)\ _
f=\q\(t — p — q)\(n — t-]-q)\(k — n — q)\ ’
Let the differential equation be
— g(x)
(21)
q = 0
(22)