I have difficulty in this problem. Show that the 1st derivative of the Legendre polynomials satisfy a self-adjoint differential equation with eigenvalue $\lambda=n(n+1)-2$.
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Thank you for your reply. Actually, I am lost with this problem. All I know is that it is related with the Sturm-Liouville Theory. – RK Ali Oct 10 '19 at 11:33
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Can you add that to your question, including perhaps information on where the problem came from and what things you tried that failed? That allows us to help you better and increases your chances of getting a good answer. Welcome to MSE! – YiFan Tey Oct 10 '19 at 11:39
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Start with the legendre differential equation, then take the derivative of both sides. You find a constant multiple of the first derivative of a legendre polynomial on one side and sums of multiples of first, second, and third derivative of that polynomial on the other. Collect the expression for the first derivative to one side, you'll find your lambda. Express what remains on the opposite side as an operator being applied to the first derivative. You get some especially interesting results applying this to the equation for Hermite Polynomials. – TurlocTheRed Oct 11 '19 at 20:50
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thank you so much.. – RK Ali Oct 12 '19 at 13:14