(1)
|
for . The Chebyshev differential equation has regular singular points at , 1, and . It can be solved by series solution using the expansions
(2)
| |||
(3)
| |||
(4)
| |||
(5)
| |||
(6)
| |||
(7)
| |||
(8)
|
Now, plug equations (6) and (8) into the original equation (◇) to obtain
(9)
|
(10)
|
(11)
|
(12)
|
(13)
|
so
(14)
|
(15)
|
and by induction,
(16)
|
for , 3, ....
Since (14) and (15) are special cases of (16), the general recurrence relation can be written
(17)
|
for , 1, .... From this, we obtain for the even coefficients
(18)
| |||
(19)
| |||
(20)
|
and for the odd coefficients
(21)
| |||
(22)
| |||
(23)
|
The even coefficients can be given in closed form as
(24)
| |||
(25)
|
and the odd coefficients as
(26)
| |||
(27)
|
The general solution is then given by summing over all indices,
(28)
|
which can be done in closed form as
(29)
|
Performing a change of variables gives the equivalent form of the solution
(30)
| |||
(31)
|
where is a Chebyshev polynomial of the first kind and is a Chebyshev polynomial of the second kind. Another equivalent form of the solution is given by
(32)
|