Analogues of the Newton formulas for the block-symmetric polynomials on $\ell_p(\mathbb{C}^s)$

Abstract

The classical Newton formulas gives recurrent relations between algebraic bases of symmetric polynomials. They are true, of course, for symmetric polynomials on infinite-dimensional sequences Banach space. In this paper we consider block-symmetric polynomials (or MacMahon symmetric polynomials) on Banach spaces $\ell_p(\mathbb{C}^s),$ $1\le p\le \infty.$ We prove an analogue of the Newton formula for the block-symmetric polynomials for the case $p=1.$ In the case $1< p$ we have no classical elementary block-symmetric polynomials. However, we extend the obtained Newton type formula for $\ell_1(\mathbb{C}^s)$ to the case of $\ell_p(\mathbb{C}^s),$ $1< p\le \infty$ and by this way found a natural definition of elementary block-symmetric polynomials on $\ell_p(\mathbb{C}^s).$

Block-Supersymmetric Polynomials on Spaces of Absolutely Convergent Series

“…In addition, we write k ≥ q whenever k 1 ≥ q 1 , k 2 ≥ q 2 , ..., k s ≥ q s . In [18], the following generalization of Newton's formula is proved (1).…”

“…, p s were considered in [17]. Some generalizations of the Newton formulas for algebraic bases of block-symmetric polynomials were obtained in [18]. Spectra of algebras of block-symmetric polynomials and holomorphic functions and algebraic structures on the spectra were considered in [19,20].…”

Rings of Multisets and Integer Multinumbers

“…Such a ring of multisets M 0 was constructed in [6] using symmetric and supersymmetric polynomials on a Banach space (see also [7]). More details about algebras of symmetric polynomials on Banach spaces can be found in [8][9][10][11][12][13][14]. The combinatorial approach to symmetric polynomials can be found in [15].…”

Spectra of algebras of block-symmetric analytic functions of bounded type

“…Note that such kinds of algebras are much more complicated and in the general case have no algebraic basis (see e.g. [21,22,[24][25][26]37]). Note that if dim X < ∞, then block-symmetric polynomials are investigated in the classical theory of invariants and combinatorics [18,32,36].…”

“…i ) ∈ C s . The connection between the basis of power block-symmetric polynomials and the basis of elementary block-symmetric polynomials is given by an analogue of the Newton formula [22,23].…”