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- W1890354621 abstract "Given a sequence ${Q_n(x)}_{n=0}^{infty}$ of symmetric orthogonal polynomials, defined by a recurrence formula $Q_n(x)=nu_ncdot xcdot Q_{n-1}(x)-(nu_n-1)cdot Q_{n-2}(x)$ with integer $nu_i$'s satisfying $nu_igeq 2$, we construct a sequence of nested Eulerian posets whose $ce$-index is a non-commutative generalization of these polynomials. Using spherical shellings and direct calculations of the $cd$-coefficients of the associated Eulerian posets we obtain two new proofs for a bound on the true interval of orthogonality of ${Q_n(x)}_{n=0}^{infty}$. Either argument can replace the use of the theory of chain sequences. Our $cd$-index calculations allow us to represent the orthogonal polynomials as an explicit positive combination of terms of the form $x^{n-2r}(x^2-1)^r$. Both proofs may be extended to the case when the $nu_i$'s are not integers and the second proof is still valid when only $nu_i>1$ is required. The construction provides a new limited testing ground for Stanley's non-negativity conjecture for Gorenstein$^*$ posets, and suggests the existence of strong links between the theory of orthogonal polynomials and flag-enumeration in Eulerian posets." @default.
- W1890354621 created "2016-06-24" @default.
- W1890354621 creator A5089863827 @default.
- W1890354621 date "2004-08-16" @default.
- W1890354621 modified "2023-10-14" @default.
- W1890354621 title "Orthogonal Polynomials Represented by $CW$-Spheres" @default.
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- W1890354621 doi "https://doi.org/10.37236/1861" @default.
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