to evaluate $\lambda_n$, at least if $p \neq 2$ when we use . \[Details of this case $p = 2$ are explained in Section \[sec:comment\]

\[lem:tjodd\] If $j$ is odd and $W$ is odd, then for $n \geq N^*$ ($N^*$ as defined in ), $$T_j(n; W) = 0.  \label{eq:toddzero}$$

By $$\begin{aligned}
T_j(n) = T_j(n; W) = \frac{1}{2\pi i} \operatorname{Trace}\int\limits_{{\partial (\mathcal{D}_n )}} \, \, (z - z_n) U_j(z) \, dz \label{eq:tjdefrep}\\
\intertext{where}
U_j = R^0 W R^0 W \dotsb W R^0 \label{eq:ujdefrep}\end{aligned}$$ with $j$ “letters” $W$ and $j + 1$ “letters” $R^0$ in this “word” $U$. All these operators are of trace class \[see Corollary \[cor:tjnorm\]\] so $\operatorname{Trace}U_j$ is a sum of the diagonal elements $(U_j)_{mm}$ which in turn are sums of matrix elements $\displaystyle \sum_g u(g)$, where $g = {\left\lbrace g_1, g_2, \dotsc, g_{j-1} \right\rbrace} \in {\mathbb{Z}}_+^{j-1}$ and

$$u(g) = \frac{1}{z - z_m} \cdot W(m, g_1) \cdot \frac{1}{z - z_{g_1}} \cdot W(g_1, g_2) \cdot \frac{1}{z - z_{g_2}} \dotsb W(g_{j-1}, m) \cdot \frac{1}{z - z_m} \label{eq:ugdef}$$

If we put $g_0 = g_j = m$ we have $$\sum_{t = 0}^{j-1} (g_{t + 1} - g_t) = g_j - g_0 = 0 \label{eq:gsum}$$ The sum of all these differences in is even (zero), 