We consider the set of Stieltjes moment sequences, for which every positive power is again a Stieltjes moment sequence, we and prove an integral representation of the logarithm of the moment sequence in analogy to the Lévy-Khinchin representation. We use the result to construct product convolution semigroups with moments of all orders and to calculate their Mellin transforms. As an application we construct a positive generating function for the orthonormal Hermite polynomials.
This paper gives via Stieltjes transform a complete description of the solution set of a matricial truncated Stieltjes-type power moment problem in the non-degenerate and degenerate cases. The approach is based on the Schur type algorithm which was worked out in the papers [arXiv:1604.07240, arXiv:1604.07629]. Furthermore, the subset of parameters is determined which corresponds to another truncated matricial Stieltjes-type moment problem.
Nondegenerate truncated indefinite Stieltjes moment problem in the class $\mathbf{N}_κ^{k}$ of generalized Stieltjes functions is considered. To describe the set of solutions of this problem we apply the Schur step-by-step algorythm, which leads to the expansion of these solutions in generalized Stieltjes continuous fractions studied recently in \cite{DK15}. Explicit formula for the resolvent matrix in terms of generalized Stieltjes polynomials is found.
This paper aims at finding conditions on a Hamburger or Stieltjes moment sequence, under which the change of at most a finite number of its entries produces another sequence of the same type. It turns out that a moment sequence allows all small enough variations of this kind precisely when it is indeterminate. We also show that a determinate moment sequence has the finite index of determinacy if and only if the corresponding finite number of its entries can be changed in a certain way.
We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of $q$-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a co…
The paper gives a parametrization of the solution set of a matricial Stieltjes-type truncated power moment problem in the non-degenerate and degenerate cases. The key role plays the solution of the corresponding system of Potapov's fundamental matrix inequalities.
Stieltjes moment sequences $\{a_n\}_{n=0}^\infty$ whose $\varkappa\,$th roots $\{\sqrtk{a_n}\}_{n=0}^\infty$ are Stieltjes moment sequences are studied ($\varkappa$ is a fixed integer greater than or equal to 2). A formula connecting the closed supports of representing measures of $\{a_n\}_{n=0}^\infty$ and $\{\sqrtk{a_n}\}_{n=0}^\infty$ is established. The relationship between the holes of the supports of these measures is investigated. The set of all pairs $(M,N)$ of positive integers for which there exists a Stieltjes moment sequence whose square root is a Stieltjes moment sequence and both…
Full indefinite Stieltjes moment problem is studied via the step-by-step Schur algorithm. Naturally associated with indefinite Stieltjes moment problem are generalized Stieltjes continued fraction and a system of difference equations, which, in turn, lead to factorization of resolvent matrices of indefinite Stieltjes moment problem. A criterion for such a problem to be indeterminate in terms of continued fraction is found and a complete description of its solutions is given in the indeterminate case. Explicit formulae for diagonal and sub-diagonal Padé approximants for formal power series corr…
The Stieltjes moment problem is studied in the framework of general Gelfand-Shilov spaces defined via weight sequences. We characterize the injectivity and surjectivity of the Stieltjes moment mapping, sending a function to its sequence of moments, in terms of growth conditions for the defining weight sequence. Finally, a related moment problem at the origin is studied.
We obtain a new multiplicative decomposition of the resolvent matrix of the truncated Hausdorff matrix moment (THMM) problem in the case of an odd and even number of moments via new Dyukarev-Stieltjes matrix (DSM) parameters. Explicit interrelations between new DSM parameters and orthogonal matrix polynomials on a finite interval $[a,b]$, as well as the Schur complements of the block Hankel matrices constructed through the moments of the THMM problem are given. Additionally, the extremal solutions of the THMM problem are represented via matrix continued fractions in terms of the DSM parameters…
We characterize the surjectivity and the existence of a continuous linear right inverse of the Stieltjes moment mapping on Gelfand-Shilov spaces, both of Beurling and Roumieu type, in terms of their defining weight sequence. As a corollary, we obtain some new results about the Borel-Ritt problem in spaces of ultraholomorphic functions on the upper half-plane.
I give a combinatorial interpretation of the multiple Laguerre polynomials of the first kind of type II, generalizing the digraph model found by Foata and Strehl for the ordinary Laguerre polynomials. I also give an explicit integral representation for these polynomials, which shows that they form a multidimensional Stieltjes moment sequence whenever $x \le 0$.
We give (necessary and sufficient) conditions over a sequence $\left\{ f_{n}\right\} _{n=0}^{\infty}$ of functions under which every generalized Stieltjes moment problem \[ \int_{0}^{\infty} f_{n}(x)ϕ(x)\mathrm{d} x=a_{n}, \ \ \ n\in\mathbb{N}, \] has solutions $ϕ\in\mathcal{S}(\mathbb{R})$ with $\operatorname*{supp} ϕ\subseteq[0,\infty)$. Furthermore, we consider more general problems of this kind for measure or distribution sequences $\left\{ f_{n}\right\} _{n=0}^{\infty}$. We also study vector moment problems with values in Frechet spaces and multidimensional moment problems.
The Stieltjes-Wigert polynomials, which correspond to an indeterminate moment problem on the positive half-line, are eigenfunctions of a second order q-difference operator. We consider the orthogonality measures for which the difference operator is symmetric in the corresponding weighted $L^2$-spaces. These measures are exactly the solutions to the q-Pearson equation.In the case of discrete and absolutely continuous measures the difference operator is essentially self-adjoint, and the corresponding spectral decomposition is given explicitly. In particular, we find an orthogonal set of q-Bessel…
We define an infinite sequence of generalizations, parametrized by an integer $m \ge 1$, of the Stieltjes--Rogers and Thron--Rogers polynomials; they arise as the power-series expansions of some branched continued fractions, and as the generating polynomials for $m$-Dyck and $m$-Schröder paths with height-dependent weights. We prove that all of these sequences of polynomials are coefficientwise Hankel-totally positive, jointly in all the (infinitely many) indeterminates. We then apply this theory to prove the coefficientwise Hankel-total positivity for combinatorially interesting sequences of …
Given a differential operator T=\sum_{i=1}^k Q_i(z)d^i/dz^i where each Q_i(z) is a polynomial define r=max_i deg(Q_i(z)-i). Assuming that r is nonnegative we consider the following multiparameter spectral problem: for each positive integer n find all polynomials V(z) of degree at most r such that the equation T(S(z))+V(z)S(z)=0 has a polynomial solution S(z) of degree n. We calculate for any converging sequence of normalized polynomials V_j(z) the root-counting measure of the corresponding sequence of polynomials S_j(z).
We show that F_a(x)=\frac{\ln Γ(x+1)}{x\ln(ax)} is a Pick function for a\ge 1 and find its integral representation. We also consider the function f(x)=(\frac{π^{x/2}}{Γ(1+x/2)})^{1/(x\ln x)} and show that \ln f(x+1) is a Stieltjes function and that f(x+1) is completely monotonic on (0,\infty). In particular f(n)=Ω_n^{1/(n\ln n)},n\ge 2 is a Hausdorff moment sequence. Here Ω_n is the volume of the unit ball in Euclidean n-space
The generalized Stieltjes--Wigert polynomials depending on parameters 0\le p<1 and 0<q<1 are discussed. By removing the mass at zero of the N-extremal solution concentrated in the zeros of the D-function from the Nevanlinna parametrization, we obtain a discrete measure μ^M which is uniquely determined by its moments. We calculate the coefficients of the corresponding orthonormal polynomials (P^M_n). As noticed by Chihara, these polynomials are the shell polynomials corresponding to the maximal parameter sequence for a certain chain sequence. We also find the minimal parameter sequence, as well…
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.
In this paper we study the linear functional $S$ on complex polynomials which is associated to a bounded complex Jacobi matrix $J$. The associated moment problem is considered: find a positive Borel measure $μ$ on $\mathbb{C}$ subject to conditions $\int z^n dμ= s_n$, where $s_n$ are prescribed complex numbers (moments). This moment problem may be viewed as an extension of the Stieltjes and Hamburger moment problems to the complex plane. Sufficient conditions for the solvability of the moment problem are provided. As a corollary, we obtain conditions for the existence of an integral representa…
The main goal of this paper is to achieve a parametrization of the solution set of the truncated matricial Hausdorff moment problem in the non-degenerate and degenerate situation. We treat the even and the odd cases simultaneously. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schur-type algorithms, namely an algebraic one and a function-theoretic one. The algebraic version, worked out in our former paper arXiv:1908.05115, is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and discussion …
Let H_N=(s_{n+m}),n,m\le N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue lambda_N of H_N. It is proved that lambda_N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of lambda_N can be arbitrarily slow or arbitrarily fast. In the indeterminate case, where lambda_N is known to be bounded below by a positive constant, we prove that the limit of the n'th smallest eigenvalue of H_N for N tending to infinity tends rapidly to infin…
We consider "Taylor domination" property for an analytic function $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k},$ in the complex disk $D_R$, which is an inequality of the form \[ |a_{k}|R^{k}\leq C\ \max_{i=0,\dots,N}\ |a_{i}|R^{i}, \ k \geq N+1. \] This property is closely related to the classical notion of "valency" of $f$ in $D_R$. For $f$ - rational function we show that Taylor domination is essentially equivalent to a well-known and widely used Turán's inequality on the sums of powers. Next we consider linear recurrence relations of the Poincaré type \[ a_{k}=\sum_{j=1}^{d}[c_{j}+ψ_{j}(k)]a_{k-j},\…
We consider symmetric Jacobi operators with recurrence coefficients such that the corresponding difference equation is in the limit circle case. Equivalently, this means that the associated moment problem is indeterminate. Our main goal is to find a representation for the resolvents of self-adjoint realizations $J$ of such Jacobi operators. This representation implies the classical Nevanlinna formula for the Cauchy-Stieltjes transforms of the spectral measures of the operators $J$. We also efficiently describe domains of the operators $J$ in terms of boundary conditions at infinity.
In this paper we study moment sequences of matrix-valued measures on compact intervals. A complete parametrization of such sequences is obtained via a symmetric version of matricial canonical moments. Furthermore, distinguished extensions of finite moment sequences are characterized in this framework. The results are applied to the underlying matrix-valued measures, generalizing some results from the scalar theory of canonical moments.
