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Find a power series solution about $x_0=0$ for the Chebyshev differential equation $$(1-x^2)y''-xy'+n^2 y=0,$$ as a function of of the integer $n$. Show that the solutions form a terminating expansion for each value of $n$. What is the orthogonality relationship for these polynomials?

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Welcome to math.stackexchange! In order to make the answers helpful for you, you should tell us what you tried and where you are stuck. –  Davide Giraudo Sep 27 '12 at 20:00
Here is a detailed solution. –  Mhenni Benghorbal Sep 27 '12 at 20:14
@DavideGiraudo I think I solved for the power series expansion first and got this as my recurrence relation. 0=a_(n+2) (n+2)(n+1)+(h^2-n^2 ) a_n Not sure where to go from there as all my coefficients turned out really wild. –  Charles Sep 27 '12 at 20:25
What is $h$ here? –  Davide Giraudo Sep 27 '12 at 20:27
sorry to avoid confusing myself, I changed the n in the primary equation to an h –  Charles Sep 27 '12 at 20:29
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