Finite Products of three terms in term of polynomials

Question

It is known that $$\prod_{i = 1}^n (1 + kx_i) = \sum_{j = 0}^ne_j(x_1, x_2, \ldots, x_n)k^j,$$ where $e_j$ is an elementary symmetric polynomial.

What about this product $$\prod_{i = 1}^n (1 - a_i + kx_i) ?$$ Can we expand it in terms of some well-known polynomials (maybe more than one type of polynomials)?

Answer 1

Given that $a_i\ne 1, \ 1\leq i\leq n$ we can bring it into a shape similar to the original one.

We have \begin{align*} \prod_{i=1}^n\left(1-a_i+kx_i\right)&=\prod_{i=1}^n\left(1-a_i\right)\,\prod_{i=1}^n\left(1+k\,\frac{x_i}{1-a_i}\right)\\ &=\prod_{i=1}^n\left(1-a_i\right)\,\sum_{j=1}^n e_j\left(\frac{x_1}{1-a_1},\cdots,\frac{x_n}{1-a_n}\right)k^j \end{align*}