Symmetric Functions and Macdonald Polynomials: Coalgebra Structure and Kawanaka Identity

This work consists of two independent parts. The first part deals with deformations of the usual basis of symmetric functions using techniques of umbral calculus. As main result the author obtains a characterization of all possible bases for the ring of symmetric functions for which the Littlewood--Richardson coefficients arise as structure coefficients. The second part solves a ten years old conjecture concerning Macdonald polynomials: the Kawanaka Macdonald polynomial conjecture.