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Sequence of polynomials

Source: 2023 Taiwan TST Round 2 Independent Study 1-N

April 5, 2023
Taiwannumber theoryalgebrapolynomial

Problem Statement

Let fnf_n be a polynomial with real coefficients for all n∈Zn \in \mathbb{Z}. Suppose that f_n(k) = f_{n+k}(k)   n, k \in \mathbb{Z}. (a) Does fn=fmf_n = f_m necessarily hold for all m,n∈Zm,n \in \mathbb{Z}? (b) If furthermore fnf_n is a polynomial with integer coefficients for all n∈Zn \in\mathbb{Z}, does fn=fmf_n = f_m necessarily hold for all m,n∈Zm, n \in\mathbb{Z}?
Proposed by usjl
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