as $N=\infty$ counterpart for the cocycle identity (\[COC\]). It is verified by $\Delta_\infty$ but, as with eq.(\[frinf\]) before, we have no idea about other possible solutions.

The $N=\infty$ sum rules
------------------------

The Taylor series of both sides of eq.(\[COBI\]) can be expressed with the help of eq.(\[DFr\]), giving $$\label{SRG}
2 \sum_{n=1}^\infty Z^{\rm P}(n) (- \lambda)^{n} \equiv 
\exp \Bigl [ \sum_{m=1}^\infty {Z(m) \over m} (- \lambda)^m  \Bigr ] -1
\equiv \sum_{r=1}^\infty {1 \over r!}
\Bigl [ \sum_{m=1}^\infty {Z(m) \over m} (- \lambda)^m  \Bigr ] ^r.$$ As in the finite-$N$ case (eq.(\[SRGN\])), this acts as a generating identity: the identification of each power $\lambda^n$ in eq.(\[SRG\]) yields a sum rule of order $n$, which here expresses the combination $2 Z^{\rm P}(n)-Z(n)/n$ in terms of the lower $Z(m)$, as $$2 Z^{\rm P}(1)-Z(1) = 0, \qquad 
2 Z^{\rm P}(2)-Z(2)/2 = Z(1)^2/2, \qquad (\mbox{etc.}).$$ The regular behaviour of eq.(\[Ffr\]) implies that these sum rules too must be the limits of their finite-$N$ counterparts (\[SR\]) and also be verified by $Z_\infty^\pm(s)$. (Due to the special form (\[Zinf\]) of $Z_\infty^\pm(s)$, these rules amount to equating each Bernoulli number $B_{2n}$ to a certain polynomial in its predecessors.)

In conclusion, this case provides a testing ground for ideas and 