@article{Kravtsiv_2020, title={Analogues of the Newton formulas for the block-symmetric polynomials on $\ell_p(\mathbb{C}^s)$}, volume={12}, url={https://journals.pnu.edu.ua/index.php/cmp/article/view/3864}, DOI={10.15330/cmp.12.1.17-22}, abstractNote={<p>The classical Newton formulas gives recurrent relations between algebraic bases of symmetric polynomials. They are true, of course, for symmetric polynomials on infinite-dimensional Banach sequence spaces.</p> <p>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 in this way we found a natural definition of elementary block-symmetric polynomials on $\ell_p(\mathbb{C}^s).$</p>}, number={1}, journal={Carpathian Mathematical Publications}, author={Kravtsiv, V.V.}, year={2020}, month={Jun.}, pages={17-22} }