Analogues of the Newton formulas for the block-symmetric polynomials on $\ell_p(\mathbb{C}^s)$
Copyright policy )Analogues of the Newton formulas for the block-symmetric polynomials on $\ell_p(\mathbb{C}^s)$
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 $\ell_p(\mathbb{C}^s).$
Newton's formula, Spaces of differentiable or holomorphic functions on infinite-dimensional spaces, newton's formula, symmetric polynomials, algebraic basis, MacMahon polynomials, block-symmetric polynomials, Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, (Spaces of) multilinear mappings, polynomials, QA1-939, Topological linear spaces of continuous, differentiable or analytic functions, Mathematics
Newton's formula, Spaces of differentiable or holomorphic functions on infinite-dimensional spaces, newton's formula, symmetric polynomials, algebraic basis, MacMahon polynomials, block-symmetric polynomials, Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, (Spaces of) multilinear mappings, polynomials, QA1-939, Topological linear spaces of continuous, differentiable or analytic functions, Mathematics
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