Coefficient of x^k in product of two polynomials with dependent degrees

Question

let $n>\ell$ be two numbers. I would like to find an expression for the coefficient $d_k$ of $x^k$ in the product

$\sum\limits_{j=1}^{\ell-1}c_jx^j\cdot\sum\limits_{i=1}^{n-\ell-1}x^i=\sum\limits_{k=2}^{n-2}d_kx^k$

(The coefficient in the second polynomial is constant 1)

The problem is that the degrees of the two polynomials are correlated. Is there a general formula for this kind of product?

Thank you,

Richard

Answer 1

Define the $(n-3)\times (m-1)$ matrix $A$ as follows:

If $k\leq j$ or $k>j+n-m-1$ then $A_{k,j}=0$ else $A_{k,j}=1$

If $C=(c_1,.....,c_{m-1})$ and $D=(d_2,.....,d_{n-2})$ are column matrices then

$$AC=D$$

Note: $m$ is $l$