{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T14:34:02Z","timestamp":1779892442896,"version":"3.53.1"},"reference-count":41,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,2]],"date-time":"2024-02-02T00:00:00Z","timestamp":1706832000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we consider a supersymmetric version of block-symmetric polynomials on a Banach space of two-sided absolutely summing series of vectors in Cs for some positive integer s&gt;1. We describe some sequences of generators of the algebra of block-supersymmetric polynomials and algebraic relations between the generators for the finite-dimensional case and construct algebraic bases of block-supersymmetric polynomials in the infinite-dimensional case. Furthermore, we propose some consequences for algebras of block-supersymmetric analytic functions of bounded type and their spectra. Finally, we consider some special derivatives in algebras of block-symmetric and block-supersymmetric analytic functions and find related Appell-type sequences of polynomials.<\/jats:p>","DOI":"10.3390\/sym16020179","type":"journal-article","created":{"date-parts":[[2024,2,2]],"date-time":"2024-02-02T09:24:08Z","timestamp":1706865848000},"page":"179","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Block-Supersymmetric Polynomials on Spaces of Absolutely Convergent Series"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7702-7071","authenticated-orcid":false,"given":"Viktoriia","family":"Kravtsiv","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., 76018 Ivano-Frankivsk, Ukraine"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,2]]},"reference":[{"key":"ref_1","unstructured":"Weyl, H. (1973). The Classical Group: Their Invariants and Representations, Princeton University Press."},{"key":"ref_2","unstructured":"Kraft, H., and Procesi, C. (1996). Classical Invariant Theory: A Primer, University of Basel. Available online: https:\/\/dmi.unibas.ch\/fileadmin\/user_upload\/dmi\/Personen\/Kraft_Hanspeter\/Classical_Invariant_Theory.pdf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1070\/SM1973v021n02ABEH002016","article-title":"On polynomial approximation of functions on Hilbert space","volume":"21","author":"Nemirovskii","year":"1973","journal-title":"Mat. USSR-Sb."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1112\/S0024610799007164","article-title":"Symmetric polynomials on rearrangement-invariant function spaces","volume":"59","author":"Gonzalo","year":"1999","journal-title":"J. Lond. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1017\/S0308210516000287","article-title":"The algebra of symmetric analytic functions on L\u221e","volume":"147","author":"Galindo","year":"2017","journal-title":"Proc. R. Soc. Edinb. Sect. A Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1007\/s13398-020-00791-w","article-title":"Analytic structure on the spectrum of the algebra of symmetric analytic functions on L\u221e","volume":"114","author":"Galindo","year":"2020","journal-title":"RACSAM"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"207","DOI":"10.5565\/PUBLMAT6612209","article-title":"Group-invariant separating polynomials on a Banach space","volume":"66","author":"Jung","year":"2022","journal-title":"Publ. Mat."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Jawad, F., and Zagorodnyuk, A. (2019). Supersymmetric polynomials on the space of absolutely convergent series. Symmetry, 11.","DOI":"10.3390\/sym11091111"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Chernega, I., and Zagorodnyuk, A. (2022). Supersymmetric Polynomials and a Ring of Multisets of a Banach Algebra. Axioms, 11.","DOI":"10.3390\/axioms11100511"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"683","DOI":"10.3390\/quantum5040043","article-title":"Applications of Supersymmetric Polynomials in Statistical Quantum Physics","volume":"5","author":"Chernega","year":"2023","journal-title":"Quantum Rep."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chopyuk, Y., Vasylyshyn, T., and Zagorodnyuk, A. (2022). Rings of Multisets and Integer Multinumbers. Mathematics, 10.","DOI":"10.3390\/math10050778"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.jalgebra.2018.10.003","article-title":"On rings of supersymmetric polynomials","volume":"517","author":"Sergeev","year":"2019","journal-title":"J. Algebra"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/0021-8693(85)90115-2","article-title":"A characterization of supersymmetric polynomials","volume":"95","author":"Stembridge","year":"1985","journal-title":"J. Algebra"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1006\/jcta.2001.3186","article-title":"MacMahon symmetric functions, the partition lattice, and Young subgroups","volume":"96","author":"Rosas","year":"2001","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_15","first-page":"4947925","article-title":"On algebraic basis of the algebra of symmetric polynomials on \u2113p(\u2102n)","volume":"2017","author":"Kravtsiv","year":"2017","journal-title":"J. Funct. Spaces"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5","DOI":"10.15330\/cmp.12.1.5-16","article-title":"Symmetric functions on spaces \u2113p(\u211dn) and \u2113p(\u2102n)","volume":"12","author":"Vasylyshyn","year":"2020","journal-title":"Carpath. Math. Publ."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Bandura, A., Kravtsiv, V., and Vasylyshyn, T. (2022). Algebraic Basis of the Algebra of All Symmetric Continuous Polynomials on the Cartesian Product of \u2113p-Spaces. Axioms, 11.","DOI":"10.3390\/axioms11020041"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"17","DOI":"10.15330\/cmp.12.1.17-22","article-title":"Analogues of the Newton formulas for the block-symmetric polynomials","volume":"12","author":"Kravtsiv","year":"2020","journal-title":"Carpath. Math. Publ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/s10958-020-04734-z","article-title":"Multiplicative convolution on the algebra of block-symmetric analytic functions","volume":"246","author":"Zagorodnyuk","year":"2020","journal-title":"J. Math. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"69","DOI":"10.30970\/ms.58.1.69-81","article-title":"Spectra of algebras of block-symmetric analytic functions of bounded type","volume":"58","author":"Kravtsiv","year":"2022","journal-title":"Mat. Stud."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"206","DOI":"10.30970\/ms.53.2.206-211","article-title":"Zeros of block-symmetric polynomials on Banach spaces","volume":"53","author":"Kravtsiv","year":"2020","journal-title":"Mat. Stud."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Macdonald, I.G. (1997). Symmetric Functions and Orthogonal Polynomials, AMS.","DOI":"10.1090\/ulect\/012"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"030010","DOI":"10.1063\/5.0115680","article-title":"Generating elements of the algebra of block-symmetric polynomials on the product of Banach spaces \u2102s","volume":"2483","author":"Kravtsiv","year":"2022","journal-title":"AIP Conf. Proc."},{"key":"ref_24","first-page":"51","article-title":"Spectra of algebras of analytic functions on a Banach space","volume":"415","author":"Aron","year":"1991","journal-title":"J. Reine Angew. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"673","DOI":"10.4153\/CJM-1995-035-1","article-title":"Weak-star continuous analytic functions","volume":"47","author":"Aron","year":"1995","journal-title":"Can. J. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1090\/S0002-9947-96-01553-X","article-title":"Regularity and algebras of analytic functions in infinite dimensions","volume":"348","author":"Aron","year":"1996","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1017\/S0013091509001655","article-title":"Some algebras of symmetric analytic functions and their spectra","volume":"55","author":"Chernega","year":"2012","journal-title":"Proc. Edinb. Math. Soc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/j.jmaa.2012.04.087","article-title":"The convolution operation on the spectra of algebras of symmetric analytic functions","volume":"395","author":"Chernega","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Baziv, N., and Zagorodnyuk, A. (2023). Analytic Invariants of Semidirect Products of Symmetric Groups on Banach Spaces. Symmetry, 15.","DOI":"10.3390\/sym15122117"},{"key":"ref_30","first-page":"381","article-title":"Continuous symmetric 3-homogeneous polynomials on spaces of Lebesgue measurable essentially bounded functions","volume":"24","author":"Vasylyshyn","year":"2018","journal-title":"Methods Funct. Anal. Topol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"125977","DOI":"10.1016\/j.jmaa.2021.125977","article-title":"Symmetric analytic functions on the Cartesian power of the complex Banach space of Lebesgue measurable essentially bounded functions on [0, 1]","volume":"509","author":"Vasylyshyn","year":"2022","journal-title":"J. Math. Anal. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1112\/blms\/bdw043","article-title":"Group-symmetric holomorphic functions on a Banach space","volume":"48","author":"Aron","year":"2016","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"104","DOI":"10.15330\/cmp.15.1.104-119","article-title":"Spectra of Algebras of Analytic Functions, Generated by Sequences of Polynomials on Banach Spaces, and Operations on Spectra","volume":"15","author":"Vasylyshyn","year":"2023","journal-title":"Carpath. Math. Publ."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"204","DOI":"10.15330\/ms.50.2.204-210","article-title":"Note on separately symmetric polynomials on the Cartesian product of \u21131","volume":"50","author":"Jawad","year":"2018","journal-title":"Mat. Stud."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"604","DOI":"10.37394\/23206.2022.21.69","article-title":"New results for degenerated generalized Apostol\u2013Bernoulli, Apostol\u2013Euler and Apostol\u2013Genocchi polynomials","volume":"21","author":"Cesarano","year":"2022","journal-title":"WSEAS Trans. Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"354","DOI":"10.15330\/cmp.14.2.354-363","article-title":"Some new classes of degenerated generalized Apostol\u2013Bernoulli, Apostol\u2013Euler and Apostol\u2013Genocchi polynomials","volume":"14","author":"Cesarano","year":"2022","journal-title":"Carpath. Math. Publ."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Bedoya, D., Cesarano, C., D\u00edaz, S., and Ram\u00edrez, W. (2023). New Classes of Degenerate Unified Polynomials. Axioms, 12.","DOI":"10.3390\/axioms12010021"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2133","DOI":"10.1090\/S0002-9939-2010-10648-2","article-title":"On multiple Appell polynomials","volume":"139","author":"Lee","year":"2011","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/s40879-019-00390-z","article-title":"Continuity and hypercyclicity of composition operators on algebras of symmetric analytic functions on Banach spaces","volume":"6","author":"Chernega","year":"2020","journal-title":"Eur. J. Math."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"360","DOI":"10.15330\/cmp.12.2.360-367","article-title":"Power Operations and Differentiations Associated with Supersymmetric Polynomials on a Banach Space","volume":"12","author":"Chernega","year":"2020","journal-title":"Carpath. Math. Publ."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1119\/1.1412643","article-title":"Partition functions and symmetric polynomials","volume":"70","author":"Schmidt","year":"2002","journal-title":"Am. J. 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