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The smallest eigenvalue of large Hankel matrices generated by a semiclassical Laguerre weight, zαe−z²+tz, where z ∈ [0,∞), α > −1, and t ∈ R, can be obtained through the asymptotics of the orthonormal polynomials Pn(z) with respect to this weight. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of the smallest eigenvalue \lambda _N, of the (N+1)\times (N+1) Hankel matrix \mathcal {M}_N=(\mu _{j+k})_{0\le j,k\le N} generated by the semi-classical Hermite weight w(z,t)=|z|^\lambda \exp \left (-z^2+tz\right), z, t \in \mathbb {R}, \lambda >-1. An asymptotic expression of the orthonormal polynomials \mathcal {P}_N(z) with the semi-classical Hermite weight w(z,t) is established as N tends to infinity. Based on the orthonormal polynomials \mathcal {P}_N(z), we obtain the specific asymptotic formulas of \lambda _{N}. [ABSTRACT FROM AUTHOR]

We give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices. To obtain these sufficient conditions, we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices. As applications, we provide a unified way to produce inequalities for many combinatorial polynomials, such as the Eulerian polynomials, Schröder polynomials, and Narayana polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study the large N behavior of the smallest eigenvalue λ N of the (N + 1) × (N + 1) Hankel matrix, H N = (μ j + k) 0 ≤ j , k ≤ N , generated by the γ dependent Jacobi weight w (z , γ) = e − γ z z α (1 − z) β , z ∈ [ 0 , 1 ] , γ ∈ R , α > − 1 , β > − 1. Applying the arguments of Szegö, Widom and Wilf, we obtain the asymptotic representation of the orthonormal polynomials P N (z) , z ∈ C ﹨ [ 0 , 1 ] , with the weight w (z , γ) = e − γ z z α (1 − z) β. Using the polynomials P N (z) , we obtain the theoretical expression of λ N , for large N. We also display the smallest eigenvalue λ N for sufficiently large N , computed numerically. [ABSTRACT FROM AUTHOR]

An asymptotic expression for the polynomials Pn(z), z ∉ (−∞, ∞), orthonormal with respect to a singularly perturbed Gaussian weight, exp(−z2−t/z2), z ∈ (−∞,∞), t > 0, is established. Based on this, the asymptotic behavior of the smallest eigenvalue of the Hankel matrix generated by the weight is described. [ABSTRACT FROM AUTHOR]

Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [J. Combin. Theory Ser. A, 74 (1996), pp. 57--70]. Schirmacher [J. Combin. Theory Ser. A, 85 (1999), pp. 127--134] extended them to q-log-concavity and almost q-log-convexity. Motivated by these, we consider the stronger properties total positivity from the Toeplitz matrix and Hankel matrix. By using exponential Riordan array methods, we give some criteria for total positivity of the triangular matrix of coefficients of the generalized cycle index polynomials, the Toeplitz matrix and Hankel matrix of the polynomial sequence in terms of the exponential formula, the logarithmic formula, and the fractional formula, respectively. Finally, we apply our criteria to some triangular arrays satisfying some recurrence relations, including Bessel triangles of two kinds and their generalizations, the Lah triangle and its generalization, the idempotent triangle, and some triangles related to binomial coefficients, rook polynomials, and Laguerre polynomials. We not only get total positivity of these lower-triangles, and q-Stieltjes moment properties and 3-q-log-convexity of their row-generating functions, but also prove that their triangular convolutions preserve the Stieltjes moment property. In particular, we solve a conjecture of Sokal on the q-Stieltjes moment property of rook polynomials. [ABSTRACT FROM AUTHOR]

In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials. [ABSTRACT FROM AUTHOR]

We provide explicit formulae for the coefficients of the order- d polynomial subresultant of ( x − α ) m and ( x − β ) n with respect to the set of Bernstein polynomials { ( x − α ) j ( x − β ) d − j , 0 ≤ j ≤ d } . They are given by hypergeometric expressions arising from determinants of binomial Hankel matrices. [ABSTRACT FROM AUTHOR]

In this paper, we elaborate on the relationship between the Hurwitz stability of matrix polynomials and matrix-valued Stieltjes functions. Our strategy is that, for a monic matrix polynomial, we associate a rational matrix-valued function with its even–odd split and then check the Hurwitz stability of the matrix polynomial by testing the Stieltjes property of the related rational matrix-valued function. On the basis of this relationship, we present matrix generalizations of a classical stability criterion by Gantmacher, Chebotarev theorem, Grommer theorem and some aspects of the modified Hermite–Biehler theorem. Our work is motivated by one of the authors' recent stability studies linked with matricial Markov parameters. [ABSTRACT FROM AUTHOR]

Abstract: Tight bounds on an elementary symmetric function are established under the assumption that the values of the elementary symmetric functions of lower orders are given. The explicit form of the inverse Hankel moment matrix leads to inequalities for moments and for elementary symmetric polynomials. [Copyright &y& Elsevier]

We give a detailed account of various connections between several classes of objects: Hankel, Hurwitz, Toeplitz, Vandermonde, and other structured matrices, Stietjes- and Jacobi-type continued fractions, Cauchy indices, moment problems, total positivity, and root localization of univariate polynomials. Along with a survey of many classical facts, we provide a number of new results. [ABSTRACT FROM AUTHOR]

It is shown that a real Hankel matrix admits an approximate block diagonalization in which the successive transformation matrices are upper triangular Toeplitz matrices. The structure of this factorization was first fully discussed in [1]. This approach is extended to obtain the quotients and the remainders appearing in the Euclidean algorithm applied to two polynomials u(x) and v(x) of degree n and m, respectively, whith m < n [ABSTRACT FROM PUBLISHER]

We consider bivariate real valued polynomials orthogonal with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between polynomials of different degrees. These formulas link the orthogonal polynomials constructed using the lexicographical ordering with those constructed using the reverse lexicographical ordering. Relations between the coefficients in the recurrence formulas are established and used to give necessary and sufficient conditions for the existence of a positive linear functional. Links to the theory of matrix orthogonal polynomials are developed as well the consequences of a zero assumption on one of the coefficients in the the recurrence formulas. [ABSTRACT FROM AUTHOR]

In this paper we associate a class of Hurwitz matrix polynomials with Stieltjes positive definite matrix sequences. This connection leads to an extension of two classical criteria of Hurwitz stability for real polynomials to matrix polynomials: tests for Hurwitz stability via positive definiteness of block-Hankel matrices built from matricial Markov parameters and via matricial Stieltjes continued fractions. We obtain further conditions for Hurwitz stability in terms of block-Hankel minors and quasiminors, which may be viewed as a weak version of the total positivity criterion. [ABSTRACT FROM AUTHOR]