Even though some important contributions to special cases were made by P. Martin-Lof, the whole circle of ideas has languished due to lack of generality. The theory presented in the present article resolves many of these problems, and provides a coherent framework for the relevant ideas. The general problem is to give conditions on the input measures such that the sequence of conclusion measures converges weakly. The resulting identities give new proofs and generalizations of the Rogers-Ramanujan identities. Two quintic transformations are given, one of which immediately proves the Rogers-Ramanujan identities without the Jacobi triple product identity.
Similar techniques lead to new transformations for unilateral and bilateral series.
The quintic transformations lead to curious identities involving primitive fifth roots of unity which are then extended to primitive pth roots of unity for odd p. 1999 Academic Press. We characterize conditions when an arbitrary function d can be represented in this way, and show how such a representation may be explicitly constructed. We also describe the extent to which the underlying tree and the edge weightings are unique in such a representation. These results generalize a recent theorem involving undirected edge assignments by an Abelian group.