Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 58

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by the corresponding Bernoullian Polynomials we obtain
the common formulae for Ð? and
One refers to y{t) as the generatrix function for i|>(/-) if
dr y(o) , s dr x(t)o
= ’"('■> —Or <22>
d or
and therefore
l(t)= »!>(«)
tn
n!
Suppose that ![)(/') is a known function, we are then able
to apply our methods to find y{t). Especially we refer to
(2:17) which indeed presents the same problem, although
with a more general aspect. Now we can make the follow-
ing substitution:—
i|>W =f(r)
dry( o)
dor
d" /(o)
dor
(22a)
where j>(x) is a conveniently chosen function, and conse-
quently
xW=j'WHio))^/(o) (22b)
Sometimes we are able to find a linear relation between
r|>W, ij»(/--f 1). • ■ • i|>(r+m), say,
cWi|>W+£(/')ii>(H-l)+ • • • • +g(r)ty(r-\-m)=o (23)
The substitution from (22) yields
a(r)
dr x(o)
dor
b(r)
dr+'x( o)
dor+1
+ •••• + g(r)
dr+n\( o)
dor+m
=o,
a (r + n)
+ • • • • +£++fl)
don
(24)