Schur type functions associated with polynomial sequences of binomial type

TL;DR

This paper introduces a new class of Schur type functions linked to polynomial sequences of binomial type, generalizing classical Schur functions and useful for describing eigenvalues in Lie algebra representations.

Contribution

It generalizes Schur functions to a broader class associated with binomial type polynomials, revealing new expansion formulas and dualities, and connecting to Lie algebra eigenvalues.

Findings

Derived new expansion formulas with duality properties

Included examples relevant to eigenvalues of Lie algebra elements

Extended classical Schur functions to polynomial sequences of binomial type

Abstract

We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur functions. This generalization satisfies some interesting expansion formulas, in which there is a curious duality. Moreover this class includes examples which are useful to describe the eigenvalues of Capelli type central elements of the universal enveloping algebras of classical Lie algebras.