Combinatorial problems from an analytical point of view
Final Report Abstract
The aim of this project was to study positivity properties that arise at the interface between analysis and combinatorics. More specifically, we were focusing on pointwise and coefficientwise total positivity for matrices built from combinatorial polynomials: in particular, for Hankel matrices and closely related combinatorial triangles. Another aim was to study analytic properties (distribution of zeros, positivity of coefficients) of certain functions emerging in combinatorial problems. A matrix of real numbers is called totally positive if all of its minors are nonnegative. This notion arose in early XXth century in analysis, and the term was coined by Schoenberg. Gantmacher and Krein shed light on spectral properties of totally positive matrices and considered the important cases of Jacobi and Hankel matrices, which are related to the Stieltjes moment problem and continued fractions. The early 1950s brought the bidiagonal factorization of totally positive matrices, as well as the factorization of functions generating Toeplitz-totally positive sequences (i.e. the sequences such that the corresponding Toeplitz matrix is totally positive). By now, the notion of total positivity found applications in classic analysis, function theory, mathematical physics, combinatorics, probability theory, group representations. Our main goal was to study positivity properties of combinatorial sequences and triangles primarily with the analytical tools. In particular, we were identifying Hankel- and Toeplitztotally positive sequences by looking at properties of their generating functions. One of the important classical tools here is the method of continued fractions. It turns out that, when the standard S- and J-fractions fail to handle an empirically Hankel-totally positive sequence, this property may in some cases be delivered by branched continued fractions. We also investigated inverse functions in the complex plane to prove that first columns of certain Riordan arrays possess a curious positivity property. We used properties of Wronski matrices to derive coefficientwise total positivity of a combinatorial triangle (a matrix of polynomials is called coefficientwise totally positive if all of its minors are polynomials with nonnegative coefficients). The method of Sturm sequences helped us to identify that rowgenerating polynomials of a certain combinatorial triangle are real-rooted, and hence their coefficients are Toeplitz-totally positive. The same method guided us toward an interesting generalization of the Eulerian triangle. Apart from the analytic methods, we exploited such tools as matrix factorizations and the Lindström–Gessel–Viennot lemma (it relates determinants to paths in directed acyclic graphs). The outcomes of this project include: (a) total positivity properties of a combinatorially interesting generalization of the Laguerre polynomials, (b) a curious improvement to a simple sufficient condition for a sequence to be Hankel-totally positive, (c) a result on perturbations of moment sequences, which helped us to clarify properties of sequences satisfying the condition from (b), (d) new families of discrete multiple orthogonal polynomials related to the classical discrete orthogonal polynomials, (e) a result on Hurwitz-stability of matrix polynomials, and (f) a study of Gauss hypergeometric functions, which yielded novel integral representations for their ratios.
Publications
- Rigidity of the Hamburger and Stieltjes moment sequences, Constr. Approx. 51 (2020), pp. 441–463
A. Dyachenko
(See online at https://doi.org/10.1007/s00365-019-09469-y) - Sokal Coefficientwise total positivity of some matrices defined by linear recurrences (2020)
X. Chen, B. Deb, A. Dyachenko, T. Gilmore, A.D.
- On generalization of classical Hurwitz stability criteria for matrix polynomials, J. Comput. Appl. Math. 383 (2021), 113113
X. Zhan, A. Dyachenko
(See online at https://doi.org/10.1016/j.cam.2020.113113) - Ratios of the Gauss hypergeometric functions with parameters shifted by integers: Part I
A. Dyachenko, D. Karp
