General formula for multiplication of n polynomials of some special form

Question

I've been playing with this on paper, but I wonder if there is a general solution or at least that is relates to some known problem. I'm trying to find

$\prod_{k = 1}^{n}[a_{k}x^{k-1} + b]$

The order of the polynomial grows quickly with n and the expression looks a bit irregular, did any of you see something similar before?

Answer 1

The special case $a_1=a_2=\ldots=a_n=a$ admits a representation as sum with q-binomial coefficients.

We obtain \begin{align*} \color{blue}{\prod_{k=1}^n\left(ax^{k-1}+b\right)} &=b^n\prod_{k=1}^n\left(\frac{a}{b}x^{k-1}+1\right)\\ &=b^n\prod_{k=0}^{n-1}\left(\frac{a}{b}x^k+1\right)\\ &=b^n\sum_{k=0}^nx^{\binom{k}{2}}\binom{n}{k}_x\left(\frac{a}{b}\right)^k\\ &\,\,\color{blue}{=\sum_{k=0}^na^kb^{n-k}x^\binom{k}{2}\binom{n}{k}_x} \end{align*}