Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 9

9
u/q d y 0
—, . ..---------r. If, therefore, m of them are known, thé
dx dx '
remaining m may be found from (10).
The solution given in (10) may as well be represented
by the following set of equations:
y- riK^i + g — f-ft)i(-*)n (n+í)
n = o (rt — n)! (t-j + s — /)! n! 'f
S — t
(s — t)\(r1 + s — t—ri)\xn (n+t)
------------—--------------%
n = o (s — t— ri)\ (rt + s —1)\ n\
+
(- l)r' xr,+s-f+1
(r-i + s — í)!
j'-.+«+1
dr' (1 — 0)'
!- t
y (»+
d ( í) x)
r. + s + l
d& ,
t = 0,1,2......(m — 1).
(11)
The equations (11) are readily derived from (10), if we
put rt = r+\—m but we could as well directly write
them down from (7). Obviously more sets of equations of
similar kind may be put down although I refrain from do-
ing so. However, in this connexion it is worth mentioning
that there exist some simple relations between the remain-
ders for different r and s. The most important ones are as
follows:
(y) = Rr+us(y)-\
= /?r,s+1 (y)
sx
(/■+s)(r + s+l)
rx
(r+s) (r+s + 1)
^r.s-l
«r-l.s
dy
dx
dy
dx
(12a)
(12b)
These formulas are readily obtained from (6) or from (1).
Further we get:
2