scholarly journals q-difference shift for a BCn-type Jackson integral arising from ‘elementary’ symmetric polynomials

2006 ◽  
Vol 204 (2) ◽  
pp. 619-646 ◽  
Author(s):  
Masahiko Ito
Keyword(s):  
Symmetric Polynomials ◽  
Jackson Integral ◽  
Elementary Symmetric Polynomials

scholarly journals Towards the Calculation of Jack Polynomials

2021 ◽  
Author(s):  
◽  
Leigh Alan Roberts
Keyword(s):  
Recent Result ◽  
Symmetric Functions ◽  
Mathematical Statistics ◽  
Jack Polynomials ◽  
Symmetric Polynomials ◽  
Geometric Progressions ◽  
Elementary Symmetric Polynomials

<p>Jack polynomials are useful in mathematical statistics, but they are awkward to calculate, and their uses have chiefly been theoretical. In this thesis a determinantal expansion of Jack polynomials in elementary symmetric polynomials is found, complementing a recent result in the literature on expansions as determinants in monomial symmetric functions. These results offer enhanced possibilities for the calculation of these polynomials, and for finding workable approximations to them. The thesis investigates the structure of the determinants concerned, finding which terms can be expected to dominate, and quantifying the sparsity of the matrices involved. Expressions are found for the elementary and monomial symmetric polynomials when the variates involved assume the form of arithmetic and geometric progressions. The latter case in particular is expected to facilitate the construction of algorithms suitable for approximating Jack polynomials.</p>

scholarly journals Hodge-Index Type Inequalities, Hyperbolic Polynomials, and Complex Hessian Equations

2019 ◽  
Author(s):  
Jian Xiao
Keyword(s):  
Type Theorem ◽  
Symmetric Polynomials ◽  
Hyperbolic Polynomials ◽  
Positivity Condition ◽  
Hessian Equations ◽  
Elementary Symmetric Polynomials ◽  
New Feature

Abstract It is noted that using complex Hessian equations and the concavity inequalities for elementary symmetric polynomials implies a generalized form of Hodge index inequality. Inspired by this result, using Gårding’s theory for hyperbolic polynomials, we obtain a mixed Hodge-index type theorem for classes of type $(1,1)$. The new feature is that this Hodge-index type theorem holds with respect to mixed polarizations in which some satisfy particular positivity condition but could be degenerate and even negative along some directions.

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