From a Polynomial Riemann Hypothesis to Alternating Sign Matrices

This paper begins with a brief discussion of a class of polynomial Riemann hypotheses, which leads to the consideration of sequences of orthogonal polynomials and 3-term recursions. The discussion further leads to higher order polynomial recursions, including 4-term recursions where orthogonality is lost. Nevertheless, we show that classical results on the nature of zeros of real orthogonal polynomials (i. e., that the zeros of pn are real and those of pn+1 interleave those of pn) may be extended to polynomial sequences satisfying certain 4-term recursions. We identify specific polynomial sequences satisfying higher order recursions that should also satisfy this classical result. As with the 3-term recursions, the 4-term recursions give rise naturally to a linear functional. In the case of 3-term recursions the zeros fall nicely into place when it is known that the functional is positive, but in the case of our 4-term recursions, we show that the functional can be positive even when there are non-real zeros among some of the polynomials. It is interesting, however, that for our 4-term recursions positivity is guaranteed when a certain real parameter C satisfies C ≥ 3, and this is exactly the condition of our result that guarantees the zeros have the aforementioned interleaving property. We conjecture the condition C ≥ 3 is also necessary. Next we used a classical determinant criterion to find exactly when the associated linear functional is positive, and we found that the Hankel determinants ∆n formed from the sequence of moments of the functional when C = 3 give rise to the initial values of the integer sequence 1, 3, 26, 646, 45885, · · · , of Alternating Sign Matrices (ASMs) with vertical symmetry. This spurred an intense interest in these moments, and we give 9 diverse characterizations of this sequence of moments. We then specify these Hankel determinants as ∗Supported in part by NSF Grant No. CCR–9821038. the electronic journal of combinatorics 8 (2001), #R36 1 Macdonald-type integrals. We also provide an an infinite class of integer sequences, each sequence of which gives the Hankel determinants ∆n of the moments. Finally we show that certain n-tuples of non-intersecting lattice paths are evaluated by a related class of special Hankel determinants. This class includes the ∆n. At the same time, ASMs with vertical symmetry can readily be identified with certain n-tuples of osculating paths. These two lattice path models appear as a natural bridge from the ASMs with vertical symmetry to Hankel determinants.