  The ring of symmetric functions $\Lambda$, with natural basis given by the
Schur functions, arise in many different areas of mathematics. For example, as
the cohomology ring of the grassmanian, and as the representation ring of the
symmetric group. One may define a coproduct on $\Lambda$ by the plethystic
addition on alphabets. In this way the ring of symmetric functions becomes a
Hopf algebra. The Littlewood--Richardson numbers may be viewed as the structure
constants for the co-product in the Schur basis. In the first part of this
thesis we show that by using a generalization of the classical umbral calculus
of Gian-Carlo Rota, one may deform the basis of Schur functions to find many
other bases for which the Littlewood--Richardson numbers as coproduct structure
constants. The Macdonald polynomials are a somewhat mysterious qt-deformation
of the Schur functions. The second part of this thesis contains a proof a
generating function identity for the Macdonald polynomials which was originally
conjectured by Kawanaka.

Combinatorics