So let $P$ be the following polynomial :
$$P(x)=\sum\limits_{n=0}^{N}\sum\limits_{k=0}^{K}\left[a_{n,k}\prod\limits_{i=1-k}^{I}(x+i)\prod\limits_{j=1-n}^{J}(x+j)\right]$$
With $N,K\in\mathbb{N}, I,J\in\mathbb{Z} \text{ such that } J>I,$ and $\left(a_{n,k}\right)_{(n,k)\in\mathbb{N}^2}$ a given double sequence.
My objective is to find the conditions on $\left(a_{n,k}\right)_{(n,k)\in\mathbb{N}^2}$ in order for $P$ to be the null polynomial : $P(x)=0$.
Now of course, $\left(\forall n,k\in\mathbb{N}, a_{n,k}=0\right)$ is a sufficient condition, but I want to know if it is also a necessary condition, that it to say that for $P$ to be the null polynomial, $\left(a_{n,k}\right)_{(n,k)\in\mathbb{N}^2}$ has to be the null sequence.
Should it be not the case (if there are $\left(a_{n,k}\right)_{(n,k)\in\mathbb{N}^2}$ that are non not the null sequence and still yield $P(x)=0$), then I'm looking at the set of thoses sequences.
If that is too hard, then I'll take the less constraining conditions on $\left(a_{n,k}\right)_{(n,k)\in\mathbb{N}^2}$ that ensures $P(x)=0$.
The most straightforward way to go would be to determine the coefficients of $P$, that it to say to find the sequence $\left(b_n\right)_{n\in\mathbb{N}}$ such that $$P(x)=\sum_{n=0}^{\deg(P)}b_nx^n$$ And then set $\forall n\in\mathbb{N},b_n=0$, and see how it constrains $\left(a_{n,k}\right)_{(n,k)\in\mathbb{N}^2}$.
In order to avoid a direct calculus of all the coefficients that looks way too tedious (if doable at all), I have thought about projecting $P$ (with an inner product) on an orthogonal polynomial basis such as the Laguerre polynomials or the Hermite Polynomials in order to have the coordinates of $P$ in that basis, and then switch them to the canonical basis $\left(1,X,...,X^{\deg(P)}\right)$ with a linear transformation.
But that might be overkill, maybe there is a solution that doesn't even involve calculating the coefficients of $P$...
How would you proceed ?