A Generalization of Schur's $P$- and $Q$-Functions

TL;DR

This paper introduces a broad generalization of Schur's $P$- and $Q$-functions linked to polynomial sequences, unifying various special cases and establishing key identities and rules for these functions.

Contribution

It presents a new framework for $P$-/$Q$-functions that encompasses several known variants and derives fundamental identities and multiplication rules within this generalized setting.

Findings

Generalized identities for $P$-/$Q$-functions

Cauchy-type identity established

Pieri-type rule derived

Abstract

We introduce and study a generalization of Schur's $P$-/$Q$-functions associated to a polynomial sequence, which can be viewed as ``Macdonald's ninth variation'' for $P$-/$Q$-functions. This variation includes as special cases Schur's $P$-/$Q$-functions, Ivanov's factorial $P$-/$Q$-functions and the $t=-1$ specialization of Hall--Littlewood functions associated to the classical root systems. We establish several identities and properties such as generalizations of Schur's original definition of Schur's $Q$-functions, Cauchy-type identity, J\'ozefiak--Pragacz--Nimmo formula for skew $Q$-functions, and Pieri-type rule for multiplication.