Rit (Vísindafélag Íslendinga) - 01.06.1951, Blaðsíða 90

[5]
90
z =x~(>,,a Ð^/a (■xp; ay)
We obtain, therefore,
from
dnz0 _ (p—n)! dny0
dxn (q—n)! dx"
Likewise with regard to (34d)
1 (\~e)Q-p-!zpy(£/x)dE
o (q-p—!)■'
(45)
(46)
(46a)
(46b)
It is evident that (45) agrees with (I8b) if we put a = L/h,
and replace p by—p, q by —p—5. It is also apparent from
(46a) if q is a positive whole number, q=m, p being frac-
dnz
tional, tbat ° = o when n>m=
' dxn
-q and z therefore a
dnv
polynomial in x of mth degree, provided that is finite
for all (positive) values of n. On the other hand, if p—q
=r, a positive whole number, we see that ip~n)!/(q- n)! is a
polynomial in n of rth degree; the A4 -series of the basal
function corresponding to (45a) consists in this case only
of r-f-1 terms, the remaining terms vanish because of
A" ) =o for n>r-\-\. The formula giving these dif-
ferences is
„ (p—o)! = (q—P—1 +n)! (p—n)!
° (q—o)! (q—P—1)/ q!
= (p—q)!(p—n)!(— 1)" = r!(p—n)!(— 1)"
q!(p—q—n)! q! (r—rí)!
(47)
which shows that the differences are zero for n>r-\-1,
provided p is fractional. However, if p is a positive integer,
m, the higher differences become also of significance ac-
cording to (47) in that we get