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Polynomial from Iran TST 2017

Source: 2017 Iran TST third exam day2 p5

April 27, 2017
IranIranian TSTpolynomialalgebra

Problem Statement

Let {ci}i=0∞\left \{ c_i \right \}_{i=0}^{\infty} be a sequence of non-negative real numbers with c2017>0c_{2017}>0. A sequence of polynomials is defined as P−1(x)=0 , P0(x)=1 , Pn+1(x)=xPn(x)+cnPn−1(x).P_{-1}(x)=0 \ , \ P_0(x)=1 \ , \ P_{n+1}(x)=xP_n(x)+c_nP_{n-1}(x). Prove that there doesn't exist any integer n>2017n>2017 and some real number cc such that P2n(x)=Pn(x2+c).P_{2n}(x)=P_n(x^2+c).
Proposed by Navid Safaei
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