Noncommutative Series Research Articles

Shift-plethysm, hydra continued fractions, and [formula omitted]-distinct partitions

We introduce the hydra continued fractions, as a generalization of the Rogers–Ramanujan continued fractions in the context of noncommutative series, and give them a combinatorial interpretation in terms of shift-plethystic trees. We show it is possible to express an m−1 headed hydra continued fraction as a quotient of m-distinct partition generating functions, and in its dual form as a quotient of the generating functions of compositions with contiguous differences upper bounded by m−1. We obtain new generating functions for compositions according to their local minima, for partitions with a prescribed set of rises, and for compositions with prescribed sets of contiguous differences.

Shift-plethystic trees and Rogers–Ramanujan identities

By studying non-commutative series in an infinite alphabet, we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the Rogers–Ramanujan identities. We prove that the language associated to shift-plethystic trees can be expressed as a non-commutative generalization of the Rogers–Ramanujan continued fraction. By specializing the non-commutative series to q-series, we obtain new combinatorial interpretations of the Rogers-Ramanujan identities in terms of signed integer compositions. We introduce the operation of shift-plethysm on non-commutative series and use this to obtain interesting enumerative identities involving compositions and partitions related to Rogers–Ramanujan identities.

Approximation of nonlinear dynamic systems by rational series

Given an analytic system, we compute a bilinear system of minimal dimension which approximates it up to order k (i.e. the outputs of these two systems have the same Taylor expansion up to order k). The algorithm is based on noncommutative series computation: let s be the generating series of the analytic system; then a rational series g is constructed, whose coefficients are equal to those of s, for all words of length smaller than or equal to k. These words are digitally encoded, in order to simplify the computations of the Hankel matrices of s and g. We then associate with g, a bilinear system, which is a solution to our problem. Another method may be used for computing a bilinear system which approximates a given analytic system ( S). We associate with ( S) an R-automaton of vector fields and build the truncated automaton by cancelling all the states which have the following property: the length of the shortest successful path labelled by a word that gets through this state is strictly greater than k. Then, the number of states of this truncated automaton yields the dimension (not necessarily minimal) of the state-space.