Rit (Vísindafélag Íslendinga) - 01.06.1946, Blaðsíða 6

On the other hand we recognize readily that, for n<r,
pn is unchanged by this transformation. In the original
serial relation (3) pn is therefore represented by (5a) for
n < r.
We have now shown how (1) may be obtained. How-
ever, it should be noticed that we could easily arrive at
the same result by direct computation.
It remains to give a suitable expression for the remain-
der Rrs(y). From the formula for qn, (5), we derive:
KM =
(-i )rx

r! s!
dr+s+ly0
(r-fs)! \(r+s+l)! dxr+s+l
(r+l)!s!jc dr+s+2y0
r+s+2
+
(r + s + 2)! 1! dx'
(r + 2)! s x2 dr+s+3y0
(r + s+3)! 2! dxr+s+3
(— 1)rxr+s+1 ^1
»r +- s -p 1
(r+s)!
í) (1 —0)
y($x)
d(í)x)
r + s + l
which may be verified by expanding
dr+s+iy
dí)
(6)
d({)x)r+s+l
Maclaurin series and integrating each term in that
/.j
into a
«r+,(l — ð)'d«
(r+ í)!s!
(r
t + 1)!
It is apparent from (6) that we can give such values to r
and s that the remainder R,-,s(y) becomes less than any
dr + s+Xy
chosen quantity, provided that — r+s+i ‘s finite in the in-
dr+s+ly
dx
r + s +
í+r maybe
terval x — 0 to x = x. At the lower limit