A large class of nonlocal equations and nonlocal charges for the Harry Dym hierarchy is exhibited. They are obtained from nonlocal Casimirs associated with its bi-Hamiltonian structure. The Lax representation for some of these equations is also given. * Supported by CNPq, Brazil. brunelli@fsc.ufsc.br . † gatcosta@mtm.ufsc.br .
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 extended 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.
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