A generalization of Newton's identity and Macdonald functions

Abstract

A generalization of Newton's identity on symmetric functions is given. Using the generalized Newton identity we give a unified method to show the existence of Hall-Littlewood, Jack and Macdonald polynomials. We also give a simple proof of the Jing-J\"ozefiak formula for two-row Macdonald functions.Comment: 14 page

Lax Operator for Macdonald Symmetric Functions

“…but depend on the parameter t only. They also emerge in the calculus of vertex operators [7] which underlies the results of [1,2,5,6,16].…”

“…In particular, the limit at N → ∞ of the renormalized Macdonald operator coefficient at u was considered in [9]. Other expressions for the same limit were given in [2,5].…”

Macdonald symmetric functions of rectangular shapes

“…However, we are interested in this commutative algebra treatment of Λ F in this study. In fact, Lemma 2.1 can also be proved-in [4] for example-inside this framework (without going back to those varialbles x 1 , x 2 , . .…”

Modular Macdonald functions and Generalized Newton's identity

“…The Macdonald polynomials have been characterized as the eigenfunctions of the Macdonald operator [M] and the Macdonald symmetric functions were also shown to be eigenfunctions of certain vertex operator like operators [AMOS, CW, GH, S, NS] (also see [CJ1,CJ2]). We follow the same strategy together with a new technique of Newton's identity [CJ2] in the modular cases. We will construct a graded differential operator X(z) = n X n z −n on the space Λ (m) with certain triangular property.…”