Rit (Vísindafélag Íslendinga) - 01.06.1951, Side 60

[4]
60
^ = x(öo)->(or) = x(Ao)-*(o//(o—r)!) (22c)
According to Nörlund10) the generatrix function for the
Bernoulii polynomial B(rz) is
Hence with respect to (22c)
b<;> =
Ð0
,Ðo_
-»0'
(é?ð»— \)-zÐzq (or)
rí_
(f—z)!
(eа—\yz(or~z)
r!
(r-z)!
Aóí(or-í)
which is in conformity with (16).
The generatrix function for Ap (or) can now be found in
a similar way
Ap (or) = (e^0—l)p(or)
and therefore (e* — l)p the generatrix function for Ap(or).
Further the generatrix function for
Ðp(o!/(o—/-)/)=(ln(l + Ao)),J (o!/(o—r)!)
is obviously (In(l+/))p in agreement with our findings
above.
We shall now show how we can apply our method to
obtain the chief formulae pertaining to a class of functions
that are important in certain problems.
We define the functions in question as follows:—
C" = AVo+(4)
Cn (u) = (u + A v/Jn qp(ri)
The basal function can now be written
—>
J>(jf(l + Atj/ío)) —»qp (r|) = Cn
n = o
dny xn
dxn n!
^TCn (u)
dny(x( 1—u)) xn
d(x(\—u))n n!
(27)