
(1) 
for . The Chebyshev differential equation has regular Singularities at
, 1, and . It can be solved by series solution using the expansions
Now, plug (24) into the original equation (1) to obtain



(5) 



(6) 



(7) 



(8) 



(9) 
so

(10) 

(11) 

(12) 
The first two are special cases of the third, so the general recurrence relation
is

(13) 
From this, we obtain for the Even Coefficients
and for the Odd Coefficients
So the general solution is



(20) 
If is Even, then terminates and is a Polynomial solution, whereas if is Odd, then terminates and is
a Polynomial solution. The Polynomial solutions defined here are known as Chebyshev Polynomials of the First
Kind. The definition of the Chebyshev Polynomial of the Second Kind gives a similar, but distinct, recurrence relation

(21) 
© 19969 Eric W. Weisstein
19990526