  The idea of orthogonal polynomials has been generalized in two ways to
achieve new types of polynomials: noncommutative orthogonal polynomials and
biorthogonal polynomials. This paper brings these two different generalizations
together to present a completely algebraic definition of noncommutative
biorthogonal polynomials. It then goes on to obtain recurrence relations for
some types of biorthogonal polynomials and concludes with a broad extension of
Favard's theorem.
