Given the 3-term recurrence relation $$(n+1)P_{n+1}=(2n+1)xP_n-nP_{n-1},$$ prove that $$(1-x^2)P''_n-2xP'_n+n(n+1)P_n=0.$$ I tried differentiate from both sides, but how can I get rid of the different indices?
2026-03-28 08:49:09.1774687749
Derive the Legendre differential equation given the recurrence relation
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