Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 15

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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)