2013 Canadian MO Problems/Problem 1

Problem

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial.

Solution

Let $F(x)=(x+1)P(x-1)-(x-1)P(x)$

$P(x)=\sum_{i=0}^{n}c_ix^i$

$F(x)=(x+1)\sum_{i=0}^{n}(x-1)^ic_i-(x-1)\sum_{i=0}^{n}c_ix^i$

$F(x)=\sum_{i=0}^{n}x(x-1)^ic_i+\sum_{i=0}^{n}(x-1)^ic_i-\sum_{i=0}^{n}c_ix^{i+1}+\sum_{i=0}^{n}c_ix^i$

$\sum_{i=0}^{n}(x-1)^ic_i=\sum_{j=0}^{n}\left( \sum_{i=j}^{n}(-1)^{i-j}\binom{i}{i-j}c_i \right)x^j$

$\sum_{i=0}^{n}x(x-1)^ic_i=\sum_{j=0}^{n}\left( \sum_{i=j}^{n}(-1)^{i-j}\binom{i}{i-j}c_i \right)x^{j+1}$

$F(x)=\sum_{j=0}^{n}\left( \sum_{i=j}^{n}(-1)^{i-j}\binom{i}{i-j}c_i \right)x^j+\sum_{j=0}^{n}\left( \sum_{i=j}^{n}(-1)^{i-j}\binom{i}{i-j}c_i \right)x^{j+1}-\sum_{i=0}^{n}c_ix^{i+1}+\sum_{i=0}^{n}c_ix^i$

In order for the new polynomial $F(x)$ to be a constant, all the coefficients in front of $x^i$ for $i>1$ need to be zero.

So we start by looking at the coefficient in front of $x^2$ :

$\left( c_2-c_1+\sum_{i=2}^{n}(-1)^{i-2}\binom{i}{i-2}c_i+\sum_{i=1}^{n}(-1)^{i-1}\binom{i}{i-1}c_i \right)x^2$

Since $\sum_{i=1}^{n}(-1)^{i-1}\binom{i}{i-1}c_i=c_1+\sum_{i=2}^{n}(-1)^{i-1}\binom{i}{i-1}c_i$ ,

$\left( c_2-c_1+c_1+\sum_{i=2}^{n}(-1)^{i-2}\binom{i}{i-2}c_i+\sum_{i=2}^{n}(-1)^{i-1}\binom{i}{i-1}c_i \right)x^2$

$\left( c_2+\sum_{i=2}^{n}\left((-1)^{i-2}\binom{i}{i-2}+(-1)^{i-1}\binom{i}{i-1} \right)c_i\right)x^2$

We then evaluate the term of the sum when $i=2$ :

$\left( c_2+(1-2)c_2+\sum_{i=3}^{n}\left((-1)^{i-2}\binom{i}{i-2}+(-1)^{i-1}\binom{i}{i-1} \right)c_i\right)x^2$

$\left(\sum_{i=3}^{n}\left((-1)^{i-2}\binom{i}{i-2}+(-1)^{i-1}\binom{i}{i-1} \right)c_i\right)x^2=0$

Therefore all coefficients $c_i$ for $i \ge 3$ need to be zero so that the coefficient in front of $x^2$ is zero.

That is, $c_3=c_4=c_5=\cdots =c_n$ .

Note that since $c_0$ , $c_1$ , and $c_2$ are not present in the expression before $x^2$ , they can be anything and the coefficient in front of $x^2$ is still zero.