**Orbit functions of ${\rm SU}(n)$ and Chebyshev polynomials**

*Maryna NESTERENKO $^\dag$, Jiří PATERA $^\ddag$ and Agnieszka TERESZKIEWICZ $^\S$*

$^\dag$ Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivs’ka Str., Kyiv-4, 01601 Ukraine

$^\ddag$ Centre de recherches mathématiques, Université de Montréal, C.P.6128-Centre ville, Montréal,\
 H3C3J7, Québec, Canada

$^\S$ Institute of Mathematics, University of Bialystok, Akademicka 2, PL-15-267 Bialystok, Poland $\phantom{^\S}$ E-mail: maryna@imath.kiev.ua, patera@crm.umontreal.ca, a.tereszkiewicz@uwb.edu.pl

Introduction {#Introduction}
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The history of the Chebyshev polynomials dates back over a century. Their properties and applications have been considered in many papers. We refer to [@Shahat; @Rivlin1974] as a basic reference. Studies of polynomials in more than one variable were undertaken by several authors, namely [@Dunkl; @DunnLidl1980; @EierLidl; @Koornwinder1-4; @MasonHandscomb2003; @suetin; @suetin2]. Of these, none follow the path we have laid down here.

In this paper, we demonstrate that the classical Chebyshev polynomials in one variable are naturally associated with the action of the Weyl group of ${\rm SU}(2)$, or equivalently with the action of the Weyl group $W(A_1)$ of the simple Lie algebra of type $A_1$. The association is so simple that it has been ignored so far. However, by making $W(A_1)$ the cornerstone of our rederivation of Chebyshev polynomials, we have gained insight 