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Sequence of polynomials divisible by p

Source: Germany 2011 - Problem 6

December 6, 2022

algebrapolynomialnumber theorySequenceprime

Problem Statement

Let

p>2

be a prime. Define a sequence

(Q_{n}(x))

of polynomials such that

Q_{0}(x)=1, Q_{1}(x)=x

and

Q_{n+1}(x) =xQ_{n}(x) + nQ_{n-1}(x)

for

n\geq 1.

Prove that

Q_{p}(x)-x^p

is divisible by

p

for all integers

x.

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