Consider the Legendre differential equation $$ (1-x^2) y'' - 2xy' + n(n+1)y = 0 $$ Then its solution is given by $$ y = c_1 P_n (x) + \text{an infinite series} $$ In fact $y = c_1 P_n (x) + c_2 Q_n (x) $ where $P_n$ is Legendre polynomials and $Q_n $ is Legendre function of the second kind. Here I want to prove that 'an infinite series' above can be written by $c_2 Q_n (x)$ for some constant $c_2$.
About the Legendre differential equation
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sequences-and-series
ordinary-differential-equations
special-functions
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0You should see [this](http://books.google.com/books?hl=en&id=Jj5pXGTZIKkC&pg=PA70). – 2012-07-29
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0@J.M. Thank you J.M., this proof is what I wanted. – 2012-07-29