Closed product formulas for certain \(R\)-polynomials (Q1348768)

\textit{D. Kazhdan} and \textit{G. Lusztig} [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)] introduced a family of polynomials (known as the Kazhdan-Lusztig polynomials), indexed by pairs of elements in a Coxeter group, which have applications in different contexts such as in the geometry of Schubert varietes and in representation theory. In order to calculate of these polynomials, another family of polynomials was defined, the \(R\)-polynomials (see, for example, Section 7.5 in the book of \textit{J. E. Humphreys} [Reflection groups and Coxeter groups (Cambridge Studies in Advanced Mathematics, 29. Cambridge etc.: Cambridge University Press) (1990; Zbl 0725.20028)]). In the given paper, the author gives a closed formula for certain \(R\)-polynomials valid for every Coxeter group. In particular, this result implies a conjecture due to \textit{F. Brenti} [Discrete Math. 193, No. 1-3, 93-116 (1998; Zbl 1061.05511)] about the \(R\)-polynomials of symmetric groups.