from Section 2 that $\alpha\in D(W)$ if and only if for any $v\in V,\; w\in W$, the formal series $\<\alpha,Y^{o}(v,x)w\>$ is a Laurent polynomial, i.e., a meromorphic function on the sphere $CP^{1}$ with only two possible poles at $0$ and $\infty$.)

In the following we give several equivalent definitions of $P(z)$-linear functional on $W$.

Let $\alpha\in W^{*}$. Then all the following conditions on $\alpha$ are equivalent:

\(a) $\alpha\in {\cal{D}}_{P(z)}(W)$.

\(b) For $v\in V$, there exist nonnegative integers $l$ and $k$ such that for every $w\in W$, $$\begin{aligned}
x^{l}(x-z)^{k}\<\alpha,Y^{o}(v,x)w\>\end{aligned}$$ is a (finite) polynomial in $x$.

\(c) For $v\in V$, there exist nonnegative integers $l$ and $k$ such that $$\begin{aligned}
x^{l}(x-z)^{k}Y^{*}(v,x)\alpha\in W^{*}[[x]].\end{aligned}$$

\(d) For $v\in V$, there exists a nonnegative integer $k$ such that $$\begin{aligned}
\label{econ1}
(x-z)^{k}Y^{*}(v,x)\alpha\in W^{*}((x)).\end{aligned}$$

From the definition, $\alpha\in {\cal{D}}_{P(z)}(W)$ if and only if for $v\in V$, there exist $l, k\in {\N}$ such that for every $w\in W$, the formal series $$x^{l}(x-z)^{k}\<\alpha,Y^{o}(v,x)w\>$$ absolutely converges in the domain $|x|>|z|$ to a rational function in $x$ without (finite) poles. Notice that a rational function without (finite) poles is simply a polynomial and any series that absolutely converges to a polynomial in the domain $|x|>|z|$ must be a polynomial itself. Then it 