A residue theorem for rational trigonometric sums and Verlinde's formula

Trigonometric identities and quadratic residues

In this paper we obtain some novel identities involving trigonometric functions. Let $n$ be any positive odd integer. We show that $$\sum_{r=0}^{n-1}\frac1{1+\sin2π\frac{x+r}n+\cos2π\frac{x+r}n} =\frac{(-1)^{(n-1)/2}n}{1+(-1)^{(n-1)/2}\sin 2πx+\cos 2πx}$$ for any complex number with $x+1/2,x+(-1)^{(n-1)/2}/4\not\in\mathbb Z$, and $$\sum_{j,k=0}^{n-1}\frac1{\sin 2π\frac{x+j}n+\sin2π\frac{y+k}n}=\frac{(-1)^{(n-1)/2}n^2}{\sin 2πx+\sin2πy}$$ for all complex numbers $x$ and $y$ with $x+y,x-y-1/2\not\in\mathbb Z$. We also determine the values of $\prod_{k=1}^{(p-1)/2}(1+\tanπ\frac{k^2}p)$ and $\prod…

$\textrm{SU}(N)$ Verlinde 和的结果式、递推公式与 Binet 公式

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

本文研究 SU(N) Verlinde 和，证明其生成多项式为迭代结果式，给出线性递推与 Binet 闭式，并附多组显式例子。