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P_n(x) = T(P_{n-1}(x))

Source: 2023 China South East Mathematical Olympiad Grade 11 P6 CSMO

April 6, 2024
algebrapolynomial

Problem Statement

Let R[x]R[x] be the whole set of real coefficient polynomials, and define the mapping T:R[x]→R[x]T: R[x] \to R[x] as follows: For f(x)=anxn+an−1xn−1+...+a1x+a0,f (x) = a_nx^{n} + a_{n-1}x^{n- 1} +...+ a_1x + a_0, let T(f(x))=anxn+1+an−1xn+(an+an−2)xn−1+(an−1+an−3)xn−2+...+(a2+a0)x+a1.T(f(x))=a_{n}x^{n+1} + a_{n-1}x^{n} + (a_n+a_{n-2})x^{n-1 } + (a_{n-1}+a_{n-3})x^{n-2}+...+(a_2+a_0)x+a_1. Assume P0(x)=1P_0(x)= 1, Pn(x)=T(Pn−1(x))P_n(x) = T(P_{n-1}(x)) ( n=1,2,...n=1,2,...), find the constant term of Pn(x)P_n(x).
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