This is called a recurrence relation or recursion formula. It gives
all the coeffi cients starting from
2
a onwards in terms of
0
aand
1
,a
which are considered as arbitrary constants. Thus we hav
e
20
31
42
0
53
1
(1)
,
2!
(1)(2)
,
3!
(2)(3)
43
(2)(1)(3)
,
4!
(3)(4)
54
(3)(1)(2)(4)
,etc.
5!
nn
aa
nn
aa
nn
aa
nnnn
a
nn
aa
nnnn
a
+
=−
−+
=−
−+
=−
⋅
−++
=
−+
=−
⋅
−−++
=
(6.16)
Substituting these coeffi
cients in Eq. (6.9), we obtain
0112
()()()yxayxayx=+ (6.17)
where
24
1
(2)(1)(3)
(1)
()1
2!4!
nnnn
nn
yxxx
−++
+
=−+− (6.18)
and
3
2
5
(1)(2)
()
3!
(3)(1)(2)(4)
5!
nn
yxxx
nnnn
x
−+
=−
−−++
+−
(6.19)
The tw
o series converge for
1,x< if they are non-terminating.
Since
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