Feynman Identity: a special case. II

TL;DR

This paper extends previous work on a special case of Feynman identity, deriving formulas for path signs, class counts, and their relation to graded Lie algebras, along with convergence proofs.

Contribution

It introduces new formulas for path class counts and links them to generalized Witt formulas, expanding understanding of the Feynman identity's special case.

Findings

Formulas for the number of equivalence classes of paths with positive or negative signs.

Connection established between path counts and generalized Witt formulas.

Proof of convergence for the infinite product in the identity.

Abstract

In this paper, the results of part I regarding a special case of Feynman identity are extended. The sign rule for a path in terms of data encoded by its word and formulas for the numbers of distinct equivalence classes of nonperiodic paths of given length with positive or negative sign are obtained for this case. Also, a connection is found between these numbers and the generalized Witt formula for the dimension of certain graded Lie algebras. Convergence of the infinite product in the identity is proved.