Power sum symmetric polynomial

Banach space

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Difference polynomials

Algebraic number

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Ring of symmetric functions

Polynomial

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Matrix polynomial

Block (permutation group theory)

Mathematical analysis

Macdonald polynomials

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Complete homogeneous symmetric polynomial

Symmetric polynomial

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Discrete orthogonal polynomials

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Elementary symmetric polynomial

Carpathian Mathematical Publications

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 $1&lt; p$ 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 $\ell_p(\mathbb{C}^s),$ $1&lt; p\le \infty$ and by this way found a natural definition of elementary block-symmetric polynomials on $\ell_p(\mathbb{C}^s).$

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