Rit (Vísindafélag Íslendinga) - 01.06.1951, Qupperneq 11

Actually, as will be shown later, thc first series represents
j’(2.x-) while the second one represents j'(c.v) because xÐx
may be transformed to Ðin.v.
Another difficulty arises when dealing witli functions like
x/ where we are able to find av, av, a^ etc. whereas it
is very diffieult to determine Ax/<*>. From the structure of
the function„we might surmise a hidden periodicity with
the period 1, the function being then not fit for the Sym-
bolic Calculus in its most general form where we cannot
dispense witli the use of aí'. All values of x! are known
for x being a positive integer or zero and we now assume
that the value of x! for any intermediate value of x may be
determined by ordinary interpolation. These assumptions
really underly (20) as it is set forth above. We cannot raise
any objections against
(i + a/M
or
xs t >' —
l>!
n U
n!(p il)!
Al'(.vs>
!>■'
-(!> - n)!Z—
n 0 r 0
v;r(-D"
r!(n—r)!
x" = i~~
Z—(p-i
n 0
xr (—i)n2r
r!(n—r)!
(22)
We may here replace x by ÐJx'1), whence we obtain
or

n u
r/.'.v" '•(—l)”jr
r!(n—r)!(q—r)!
x "fl('(.v")-'
n 0
p!_
(p—n)!
q!()/x)r (-l)”-r
r!(n—r)!(q—r)!
If we, now, substitute y (/' ,,.v" " for Ð'!(xq) as indicated
(q—p)!
by (19) we arrive at (20) by replacing x by \/x. The close
relationship between (19) and (20) is thereby borne out.
From (20) we get