a ${\C}[x,x^{-1}]$-valued linear function on $V$ by considering $\<w',Y(v,x)w\>$ for $v\in V$. To get a ${\C}$-valued function on $V$, we use Huang and Lepowsky’s idea to evaluate the formal variable $x$ at a nonzero complex number $z$ as in \[H3\] and \[HL0-3\], so that $w'\otimes w$ can be viewed as a linear functional on $V$ by $$\begin{aligned}
(w'\otimes w)(v)=\<w',Y(v,z)w\>\;\;\;\mbox{ for }v\in V.\end{aligned}$$ Such functionals satisfy a certain nice rationality property as described below. For a linear functional $\alpha$ on $V$, we naturally extend $\alpha$ to a linear function on $V[[x,x^{-1}]]$. Then consider the formal series $\alpha(Y(u,x)v)$ with $u,v\in V$. If $\alpha=w'\otimes w$, we have $$\alpha(Y(u,x)v)=\<w',Y(Y(u,x)v,z)w\>,$$ which by associativity converges to a rational function in the domain $0<|x|<|z|$ with only two possible (finite) poles at $0$ and $-z$. Globally, from $Y$ we obtain a linear map $F$ from $W'\otimes W$ to $V^{*}$ defined by $$\begin{aligned}
\<F(w'\otimes w), v\>=\<w',Y(v,z)w\>\end{aligned}$$ for $w'\in W',\; w\in V,\; v\in V$. This map $F$ is nothing but a $Q(z)$-intertwining map as known in \[HL1\]. Linear functionals $\alpha$ with this rationality property are called $Q(z)$-linear functions and will be studied in \[Li6\].

In this paper we shall study the so-called $P(z)$-linear functionals defined below. Notice that previously we 