we call the $n$th circle products. For homogeneous $a,b\in QO(V)$, they are defined by $$a(w)\circ_n b(w)=Res_z a(z)b(w)~\iota_{|z|>|w|}(z-w)^n-
(-1)^{|a||b|}Res_z b(w)a(z)~\iota_{|w|>|z|}(z-w)^n.$$ Here $\iota_{|z|>|w|}f(z,w)\in\mathbb{C}[[z,z^{-1},w,w^{-1}]]$ denotes the power series expansion of a rational function $f$ in the region $|z|>|w|$. We usually omit the symbol $\iota_{|z|>|w|}$ and just write $(z-w)^{-1}$ to mean the expansion in the region $|z|>|w|$, and write $-(w-z)^{-1}$ to mean the expansion in $|w|>|z|$. It is easy to check that $a(w)\circ_n b(w)$ above is a well-defined element of $QO(V)$.

The nonnegative circle products are connected through the [*operator product expansion*]{} (OPE) formula. For $a,b\in QO(V)$, we have $$\label{opeform} a(z)b(w)=\sum_{n\geq 0}a(w)\circ_n
b(w)~(z-w)^{-n-1}+:a(z)b(w):\ ,$$ which is often written as $a(z)b(w)\sim\sum_{n\geq 0}a(w)\circ_n b(w)~(z-w)^{-n-1}$, where $\sim$ means equal modulo the term $$:a(z)b(w):\ =a(z)_-b(w)\ +\ (-1)^{|a||b|} b(w)a(z)_+.$$ Here $a(z)_-=\sum_{n<0}a(n)z^{-n-1}$ and $a(z)_+=\sum_{n\geq
0}a(n)z^{-n-1}$. Note that $:a(w)b(w):$ is a well-defined element of $QO(V)$. It is called the [*Wick product*]{} of $a$ and $b$, and it coincides with $a\circ_{-1}b$. The other negative circle products are related to this by $$n!~a(z)\circ_{-n-1}b(z)=\ :(\partial^n a(z))b(z):\ ,$$ where $\partial$ denotes the formal differentiation operator $\frac{d}{dz}$. For $a_1(z),\dots ,a_k(z)\in QO(V)$, the $k$-fold iterated Wick product is defined to be $$\label{iteratedwick} :a_1(z)a_2(z)\cdots a_k(z):\ =\ :a_1(z)b(z):~,$$ where $b(z)=\ :a_2(z)\cdots a_k(z):\ $. We often omit the formal variable 