Waring-Girard formulas for block-symmetric and block-supersymmetric polynomials

This paper investigates the structure and properties of block-symmetric and block-super\-symmetric polynomials in Banach spaces. The study extends classical symmetric polynomial results to infinite-dimensional settings, particularly in sequence spaces such as $\ell_p(\mathbb{C}^s),$ $1\leq p<\infty$ and spaces of two-sided absolutely summing series of vectors in $\mathbb{C}^s$ for some positive integer $s>1.$ In this paper, we derive analogs of the Waring-Girard formulas for block-symmetric and block-supersymmetric polynomials and explore their combinatorial applications.