Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 40
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(a-\-2r)!:
(a-\-r)!r!( l+x)a+2,'+i (l + x)a+i
(aJr,2r)/l__'x\
r!(a+r)!\(\+x)2!
or
(l+x)a+; _+?- (g+2n)! I x \
1— x n!(a+rí)! 1(1+x)2 /
Applying the operator to (27) we find
(27)
•v'- (1 ! xY' f' (a+2rí)!xn
\—x ^—(n—r)!(a+n+r)!(\+x)2n ^ '
n= r
whence, as above,
¥, - a+\+2r (a+2rí)! xn
(l+x)a+/ (n—r)! (a+n+r+\)! (1+x)2n ' '
n = r
Replacing a by a—1 and dealing with (29) as 1 have above
dealt with (25), I obtain
y =
^ (a + \+2rí)!xn
7~o (l+x)-"+a
(a + 2 r)
d+yo
dxr
~r0 r!(n—r)!(a+n+r)!
(30)
Evidently (30) is an analogon to (26). The series (26) and
(30) include the arbitrary quantity a. We can put such value
on a as to let the (/re+l)th term of the series, (n=m), vanish;
the series in question then becomes an asymptotic series
and may serve to compute y, approximately.
The methods set out above to calculate infinite series
are all based on the series Ai to A7 which are equivalent
to the Basal Function. We observe that these series are
power series, i. e. one quantity at least in each term appears
as increasing powers as we go to higher terms. Sometimes
we have to deal with infinite series that do not appear as