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Results about the sequence of polynomials f_n(x)

Source: Chinese MO 1999

March 19, 2012
algebrapolynomialfunctioninductionalgebra proposed

Problem Statement

Let aa be a real number. Let (fn(x))n≥0(f_n(x))_{n\ge 0} be a sequence of polynomials such that f0(x)=1f_0(x)=1 and fn+1(x)=xfn(x)+fn(ax)f_{n+1}(x)=xf_n(x)+f_n(ax) for all non-negative integers nn. a) Prove that fn(x)=xnfn(x−1)f_n(x)=x^nf_n\left(x^{-1}\right) for all non-negative integers nn. b) Find an explicit expression for fn(x)f_n(x).
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