I recently learned about the Chebyshev's polynomial of second kind, denoted by $U_{n}(x)$, and defined by the polynomial of degree $n$ that sends $\cos(\theta)$ to $\frac{\sin(n+1)\theta}{\sin(\theta)}$. These polynomials also satisfy the second-order recurrence relation given as $$ U_{0}(x) = 1, U_{1}(x) = 2x \text{ and } U_{n}(x) = 2xU_{n-1}(x) - U_{n-2}(x) \text{ for } n\geq 2.$$
Given a natural number $N$, I am interested in finding the resultant of two polynomials $U_{m}(x) + U_{m-1}(x)$ and $U_{n}(x) + U_{n-1}(x)$, where $1 \leq m,n \leq N$.
Recently, I came across the this paper ( https://www.sciencedirect.com/science/article/pii/S0022314X15002140) that studies the resultants of $U_{m}(x)$ and $U_{n}(x)$ for arbitrary $m$ and $n$. However, this does not seem to be helpful in my problem as expected, because resultant of sum bears not much relation (at least, apriori) to the original resultant.