Home Competitions ILL 1978 Problem 47
Given the expression
P n ( x ) = 1 2 n [ ( x + x 2 − 1 ) n + ( x − x 2 − 1 ) n ] , P_n(x) =\frac{1}{2^n}\left[(x +\sqrt{x^2 - 1})^n+(x-\sqrt{x^2 - 1})^n\right], P n ( x ) = 2 n 1 [ ( x + x 2 − 1 ) n + ( x − x 2 − 1 ) n ] , ( a ) P n ( x ) (a) P_n(x) ( a ) P n ( x ) P n ( x ) − x P n − 1 ( x ) + 1 4 P n − 2 ( x ) ≡ 0. P_n(x) - xP_{n-1}(x) + \frac{1}{4}P_{n-2}(x) \equiv 0. P n ( x ) − x P n − 1 ( x ) + 4 1 P n − 2 ( x ) ≡ 0. ( b ) P n ( x ) (b) P_n(x) ( b ) P n ( x ) x x x n . n. n .
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