From a Polynomial Riemann Hypothesis to Alternating Sign Matrices

Abstract

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 $p_n$ are real and those of $p_{n+1}$ interleave those of $p_n$) 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 $C\ge 3$, and this is exactly the condition of our result that guarantees the zeros have the aforementioned interleaving property. We conjecture the condition $C\ge 3$ 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 $\Delta_n$ formed from the sequence of moments of the functional when $C = 3$ give rise to the initial values of the integer sequence $1, 3, 26, 646, 45885, \cdots,$ 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 Macdonald-type integrals. We also provide an an infinite class of integer sequences, each sequence of which gives the Hankel determinants $\Delta_n$ 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 $\Delta_n$. 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.

Advanced determinant calculus: A complement

“…In [53,Theorem 4], Egecioglu, Redmond and Ryavec prove also the following common generalisation of (5.49) and (5.50). (The first identity is the special case x = 0, while the second is the special case x = 1 of the following theorem.)…”

The Generating Function of Ternary Trees and Continued Fractions

“…Tamm's proof used the fact that Hankel determinants can be evaluated using continued fractions; the continued fraction that gives these Hankel determinants is a special case of Gauss's continued fraction for a quotient of hypergeometric series. The determinant V n was also evaluated, using a different method, by Egecioglu, Redmond, and Ryavec [6,Theorem 4], who also noted the connection with alternating sign matrices and gave several additional Hankel determinants for V n : V n = det (b i+j ) 0≤i,j≤n−1 = det (r i+j ) 0≤i,j≤n−1 = det (s i+j (u)) 0≤i,j≤n−1 , where u is arbitrary. As noted in [6,Theorem 4], s n (0) = b n , s n (1) = a n+1 , and s n (3) = r n .…”

“…The determinant V n was also evaluated, using a different method, by Egecioglu, Redmond, and Ryavec [6,Theorem 4], who also noted the connection with alternating sign matrices and gave several additional Hankel determinants for V n : V n = det (b i+j ) 0≤i,j≤n−1 = det (r i+j ) 0≤i,j≤n−1 = det (s i+j (u)) 0≤i,j≤n−1 , where u is arbitrary. As noted in [6,Theorem 4], s n (0) = b n , s n (1) = a n+1 , and s n (3) = r n . In Section 2, we describe Tamm's continued fraction method for evaluating these determinants.…”

Weight-preserving bijections between integer partitions and a class of alternating sign trapezoids

“…Osculating paths are lattice paths that neither cross nor share edges but potentially share points. The idea of describing ASMs in terms of osculating paths dates back to [BH95] and was further investigated in [Bra97], [ERR01], and [Beh08], among others.…”