Resultants, Recursive Formulas and Binet Formulas for $\text{SU}(N)$ Verlinde Sums

We study Verlinde sums $V_n(N,m)$, which for our purposes are finite sums of trigonometric functions $\{r_x\}$, associated to $\text{ SU}(N)$. In this article we prove the following results: (i) the polynomial $D_N(w):={\prod }_x(w-r_x)$ is equal to, up to an explicit and computable factor, the iterated resultant of $Q(u)=u^m-1$ with an explicit elementary polynomial $W_N$ which is independent of $n$ and $m$; (ii) for fixed $N$ and $m$, $V_n(N,m)$ satisfies a linear recursion of order at most $\left(\genfrac{}{}{0px}{}{m-1}{N-1}\right)$, together with arguments showing that one actually has a recursion of length equal to the number of distinct roots of $D_N$; (iii) for fixed $N$ and $m$, a Binet-type closed form expressing $V_n(N,m)$ as an explicit finite sum of $n$-th powers of algebraic numbers which are rescaled reciprocals of the distinct roots of $D_N(w)$ is derived. Several explicit examples are provided for $N=2,3,4,5,6$ and $8$.