  Quasi-shuffle products, introduced by the first author, have been useful in
studying multiple zeta values and some of their analogues and generalizations.
The second author, together with Kajikawa, Ohno, and Okuda, significantly
extended the definition of quasi-shuffle algebras so it could be applied to
multiple zeta q-values. This article extends some of the algebraic machinery of
the first author's original paper to the more general definition, and uses this
extension to obtain various algebraic formulas in the quasi-shuffle algebra in
a transparent way. Some applications to multiple zeta values, interpolated
multiple zeta values, multiple q-zeta values, and multiple polylogarithms are
given.
