Orthogonal Polynomials Represented by $CW$-Spheres

Given a sequence $\{Q_n(x)\}_{n=0}^{\infty}$ of symmetric orthogonal polynomials, defined by a recurrence formula $Q_n(x)=\nu_n\cdot x\cdot Q_{n-1}(x)-(\nu_n-1)\cdot Q_{n-2}(x)$ with integer $\nu_i$'s satisfying $\nu_i\geq 2$, 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.