Connections between polynomial sets: A group-theoretic framework

Abstract This paper develops a comprehensive group-theoretic framework for analyzing transformations and connections between polynomial sets. Building upon the fundamental characterization [Y. Ben Cheikh, Some structures on the polynomial sets, Dolomites Res. Notes Approx., to appear] of the space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℙ</m:mi> </m:math> {\mathbb{P}} of all polynomial sets as the Cartesian product <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℙ</m:mi> <m:mo>≅</m:mo> <m:mrow> <m:mrow> <m:msup> <m:mi mathvariant="normal">Λ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msup> <m:mo>/</m:mo> <m:mi mathvariant="script">ℛ</m:mi> </m:mrow> <m:mo>×</m:mo> <m:msub> <m:mi mathvariant="script">𝒮</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>×</m:mo> <m:msub> <m:mi mathvariant="script">𝒮</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:mrow> </m:math> {\mathbb{P}\cong\Lambda^{(-1)}/\mathscr{R}\times\mathcal{S}_{1}\times\mathcal{% S}_{0}} , we introduce and systematically study T -operators acting on this space. These operators provide a unified mechanism for connecting arbitrary polynomial sequences, while stabilizer analysis reveals the inherent symmetry properties of classical polynomial families. Our approach yields powerful new tools for addressing fundamental problems in polynomial theory, including connection coefficients, transformation classification, and the systematic study of relationships between diverse polynomial families. The framework provides both theoretical insights and practical computational methods for working with polynomial sets and their interconnections.