Two collection formulas

Abstract

Let G be a group, x,y∈G, and H is a normal closure in G of the subgroup generated by all commutators of x,y that include more than two occurrences of y. Our main result is a parametrization of the uncollected part of Hall’s collection formula for (xy)^n and two different collection formulas modulo H in an explicit form. In particular, we found explicit expressions for the exponents of the commutators in Hall’s collection formula for both known cases and some unknown cases. Also we give combinatorial identities, which simplify those expressions and help to investigate their divisibility.