Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 5

5
r+ 1 ternis on the left hand side of (3) are approximately
equal to the sum of the first s + 1 terms on the right hand
side, the correction Rrs.(y) being so small that it may be left
out of consideration. Certainly this may be obtained in vari-
ous ways. I have contrived to get this result by assuming
pr+n = 0 for n > 1, and besides this I give to the co-
efficients pv p2, p3...pr such values that
qs+l = qs+2 = qt+3 =••••= q,+r=°-
From the fact that pr+n — 0,1 conclude with regard to (4a)
that qn is a polynomial in n of degree r. Further it is ob-
vious that the roots of qn are s + 1, s + 2, s + 3, ...s + r.
Hence, with q0= 1,

M(,_ M
s+l/V s + 2/
s! (s + r— n)\
(s — rí)! (s + r)!
_JL.\
s + r
(5)
Regarding the remainder, we see that it constitutes an in-
finite series multiplied by jc'+s + 1 where the first term of
the series includes
dr+s+'y0
in stead of
y o-
The coef-
dxr+s+l
ficients q of this infinite series are wholly determined as
wili become evident. Consequently we may replace this
series by the corresponding series on the left hand side
of (3). Moving the terms thus obtained to the left hand side
of the original series, we recognize that the series is now
transformed to another series where q , = 0 for n > 1.
By application of the same reasoning as before we now
find readily
n
r+2
/-! (r + s — n)
(r—n)! (r + s)!
(5a)