  The subject of this paper are operators represented on Fock spaces whose
behavior on one level depends only on two of its neighbors. Our initial
objective was to generalize (via a common framework) the results of
arXiv:math/0702158, arXiv:0709.4334, arXiv:0812.0895, and arXiv:1003.2998,
whose constructions exhibited this behavior. We extend a number of results from
these papers to our more general setting. These include the quadratic relation
satisfied by the free cumulant generating function (actually by a variant of
it), the resolvent form of the generating function for the Wick polynomials,
and classification results for the case when the vacuum state on the operator
algebra is tracial. We are able to handle the generating functions in
infinitely many variables by considering their matrix-valued versions. Finally,
we provide norm estimates guaranteeing that these generating functions are
represented by bounded operators.
