Stieltjes Function Research Articles

A complete Bernstein function related to the fractal dimension of Pascal’s pyramid modulo a prime

Let fr(x)=log(1+rx)/log(1+x) for x>0. We prove that fr is a complete Bernstein function for 0≤r≤1 and a Stieltjes function for 1≤r. This answers a conjecture of David Bradley that fr is a Bernstein function when 0≤r≤1.

Diverse characterizations of Q-Dunkl classical orthogonal polynomials

In this paper, we present four characterizations of q-Dunkl-classical orthogonal polynomials. The first one is a T θ , q -distributional equation of Pearson type fulfilled by its associated form, where T θ , q is the q-Dunkl operator. The second is a second order linear T θ , q -difference equation. The third is a first order linear T θ , q -difference equation with polynomial coefficients satisfied by the corresponding Stieltjes function. The fourth is the so-called structure relation fulfilled by q-Dunkl-classical polynomials. Moreover, we provide an example of a non-symmetric sequence of T θ , q -classical orthogonal polynomials.

Lax-type pairs in the theory of bivariate orthogonal polynomials

Sequences of bivariate orthogonal polynomials written as vector polynomials of increasing size satisfy a couple of three term relations with matrix coefficients. In this work, introducing a time-dependent parameter, we analyze a Lax-type pair system for the coefficients of the three term relations. We also deduce several characterizations relating the Lax-type pair, the shape of the weight, Stieltjes function, moments, a differential equation for the weight, and the bidimensional Toda-type systems.

Several characterizations of third-degree semiclassical linear forms of class two appearing via cubic decomposition

ABSTRACT In this work, we consider orthogonal polynomials via cubic decomposition in the framework of the third-degree semiclassical class. Based on their third-degree character, we give a complete description, by using the formal Stieltjes function and the moments, of semiclassical linear forms of class two arising from the cubic decomposition W 3 n ( x ) = P n ( x 3 ) , n ≥ 0 . We focus our attention on the link between these forms and the strict third-degree classical forms V q k , l = J ( k + q / 3 , l − q / 3 ) , q ∈ { 1 , 2 } , k , l ∈ Z with k + l ≥ − 1 . All of them are rational transformations of the Jacobi form V = J ( − 2 / 3 , − 1 / 3 ) .

Stieltjes functions and spectral analysis in the physics of sea ice

Abstract. Polar sea ice is a critical component of Earth’s climate system. As a material, it is a multiscale composite of pure ice with temperature-dependent millimeter-scale brine inclusions, and centimeter-scale polycrystalline microstructure which is largely determined by how the ice was formed. The surface layer of the polar oceans can be viewed as a granular composite of ice floes in a sea water host, with floe sizes ranging from centimeters to tens of kilometers. A principal challenge in modeling sea ice and its role in climate is how to use information on smaller-scale structures to find the effective or homogenized properties on larger scales relevant to process studies and coarse-grained climate models. That is, how do you predict macroscopic behavior from microscopic laws, like in statistical mechanics and solid state physics? Also of great interest in climate science is the inverse problem of recovering parameters controlling small-scale processes from large-scale observations. Motivated by sea ice remote sensing, the analytic continuation method for obtaining rigorous bounds on the homogenized coefficients of two-phase composites was applied to the complex permittivity of sea ice, which is a Stieltjes function of the ratio of the permittivities of ice and brine. Integral representations for the effective parameters distill the complexities of the composite microgeometry into the spectral properties of a self-adjoint operator like the Hamiltonian in quantum physics. These techniques have been extended to polycrystalline materials, advection diffusion processes, and ocean waves in the sea ice cover. Here we discuss this powerful approach in homogenization, highlighting the spectral representations and resolvent structure of the fields that are shared by the two-component theory and its extensions. Spectral analysis of sea ice structures leads to a random matrix theory picture of percolation processes in composites, establishing parallels to Anderson localization and semiconductor physics and providing new insights into the physics of sea ice.

Truncated Hermite polynomials

We define the family of truncated Hermite polynomials , orthogonal with respect to the linear functional The connection of with Hermite and Rys polynomials is stated. The semiclassical character of as polynomials of class 2 is emphasized. As a consequence, several properties of concerning the coefficients in the three-term recurrence relation they satisfy as well as the moments and the Stieltjes function of L are given. Ladder operators associated with such a linear functional and the holonomic equation that the polynomials satisfy are deduced.

Characterization theorem for strict third-degree semiclassical forms of class one obtained via cubic decomposition

Based on their third-degree character, we give a complete description of a large family of semiclassical forms of class one. A characterization theorem for all strict third-degree semiclassical forms of class one arising via the cubic decomposition W_{3n}(x)=P_n(x^3+qx+r) , q, r\in\mathbb{C} , n\geq 0 , is stated and proved. In fact, by using the Stieltjes function and the moments of those forms, this theorem gives necessary and sufficient conditions that characterize and identify all these forms, and we present their link with the Jacobi forms \mathcal{V}_q^{k, l}=\mathcal{J}(k+q/3,l-q/3) , k+l\geq-1 , k, l \in \mathbb{Z} , q\in\{1,2\} .

Integrable differential systems for deformed Laguerre–Hahn orthogonal polynomials

Our work studies sequences of orthogonal polynomials of the Laguerre–Hahn class, whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients, which are subject to a deformation parameter t. We derive systems of differential equations and give Lax pairs, yielding nonlinear differential equations in t for the recurrence relation coefficients and Lax matrices of the orthogonal polynomials. A specialisation to a non semi-classical case obtained via a Möbius transformation of a Stieltjes function related to a deformed Jacobi weight is studied in detail, showing this system is governed by a differential equation of the Painlevé type P. The particular case of P arising here has the same four parameters as the solution found by Magnus (1995 J. Comput. Appl. Math. 57 215–37) but differs in the boundary conditions.

Volterra-Prabhakar function of distributed order and some applications

The paper studies the exact solution of two kinds of generalized Fokker–Planck equations in which the integral kernels are given either by the distributed order function k1(t)=∫01t−μ/Γ(1−μ)dμ or the distributed order Prabhakar function k2(α,γ;λ;t)=∫01eα,1−μ−γ(λ;t)dμ, where the Prabhakar function is denoted as eα,1−μ−γ(λ;t). Both of these integral kernels can be called the fading memory functions and are the Stieltjes functions. It is also shown that their Stieltjes character is enough to ensure the non-negativity of the mean square values and higher even moments. The odd moments vanish. Thus, the solution of generalized Fokker–Planck equations can be called the probability density functions. We introduce also the Volterra-Prabhakar function and its generalization which are involved in the definition of k2(α,γ;λ;t) and generated by it the probability density function p2(x,t).

Generalized Bernstein functions

A class of functions called generalized Bernstein functions is studied. The fundamental properties of this class are given and its relation to generalized Stieltjes functions via the Laplace transform is investigated. The subclass of generalized Thorin-Bernstein functions is characterized in different ways. Examples of generalized Bernstein functions include incomplete gamma functions, Lerch's transcendent and some hypergeometric functions.

Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes Function

This paper considers the computation of approximations of matrix functionals of form F(A):=vTf(A)v, where A is a large symmetric positive definite matrix, v is a vector, and f is a Stieltjes function. The functional F(A) is approximated by a rational Gauss quadrature rule with poles on the negative real axis (or part thereof) in the complex plane, and we focus on the allocation of the poles. Specifically, we propose that the poles, when considered positive point charges, be allocated to make the negative real axis (or part thereof) approximate an equipotential curve. This is easily achieved with the aid of conformal mapping.

A unified rational Krylov method for elliptic and parabolic fractional diffusion problems

SummaryWe present a unified framework to efficiently approximate solutions to fractional diffusion problems of stationary and parabolic type. After discretization, we can take the point of view that the solution is obtained by a matrix‐vector product of the form , where is the discretization matrix of the spatial operator, a prescribed vector, and a parametric function, such as a fractional power or the Mittag‐Leffler function. In the abstract framework of Stieltjes and complete Bernstein functions, to which the functions we are interested in belong to, we apply a rational Krylov method and prove uniform convergence when using poles based on Zolotarëv's minimal deviation problem. The latter are particularly suited for fractional diffusion as they allow for an efficient query of the map and do not degenerate as the fractional parameters approach zero. We also present a variety of both novel and existing pole selection strategies for which we develop a computable error certificate. Our numerical experiments comprise a detailed parameter study of space‐time fractional diffusion problems and compare the performance of the poles with the ones predicted by our certificate.

A theory of composites perspective on matrix valued Stieltjes functions

A series of physically motivated operations appearing in the study of composite materials are interpreted in terms of elementary continued fraction transforms of matrix valued, rational Stieltjes functions.

On Non-linear Characterizations of Classical Orthogonal Polynomials

Classical orthogonal polynomials are known to satisfy seven equivalent properties, namely the Pearson equation for the linear functional, the second-order differential/difference/q-differential/ divided-difference equation, the orthogonality of the derivatives, the Rodrigues formula, two types of structure relations, and the Riccati equation for the formal Stieltjes function. In this work, following previous work by Kil et al. (J Differ Equ Appl 4:145–162, 1998a; Kyungpook Math J 38:259–281, 1998b), we state and prove a non-linear characterization result for classical orthogonal polynomials on non-uniform lattices. Next, we give explicit relations for some families of these classes.

On the relation between Hurwitz stability of matrix polynomials and matrix-valued Stieltjes functions

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.

Reconstructing Stieltjes Functions from Their Approximate Values: A Search for a Needle in a Haystack

Material response of passive media to external influences is described by complex analytic functions of frequency that can always be written in terms of Stieltjes functions -- a special class of analytic functions mapping complex upper half-plane into itself. Reconstructing such functions from their experimentally measured values at specific frequencies is one of the central problems that we address in this paper. A definitive reconstruction algorithm that produces a certificate of optimality as well as a graphical representation of the uncertainty of reconstruction is proposed. Its effectiveness is demonstrated in the context of the electrochemical impedance spectroscopy.

Rational gauss quadrature rules for the approximation of matrix functionals involving stieltjes functions

Rational gauss quadrature rules for the approximation of matrix functionals involving stieltjes functions

Second degree linear forms and semiclassical forms of class one: A case study

Based on their second degree character, in this contribution we study new characterizations of families of symmetric and quasi-symmetric semiclassical linear forms of class one. In fact, by using the Stieltjes function and the moments of those forms, we give necessary and sufficient conditions for a regular form to be at the same time of the second degree, symmetric (resp. quasi-symmetric) and semiclassical of class one. We focus our attention on the link between these forms and the Jacobi forms Tp,q = J(p-1/2, q-1/2), p,q ? Z, p+q ? 0. All of them are rational transformations of the first kind Chebychev form T = J (-1/2,-1/2). Finally, we study a family of second degree linear forms which are semiclassical of class one and are not included in the above families.

Distributions of zeros and poles of -point Padé approximants to complex-symmetric functions defined at complex points

UDC 517.5 The knowledge of the location of zeros and poles Padé and -point Padé approximations to a given function provides much valuable information about the function being studied.In general PAs reproduce the exact zeros and poles of considered function, but, unfortunately, some spurious zeros and poles appear randomly.Then, it is clear that the control of the position of poles and zeros becomes essential for applications of Padé approximation method.The numerical examples included in the paper show how necessary for the convergence of PA is the knowledge of the position of their zeros and poles.We relate our research of localization of poles and zeros of PA and NPA in the case of Stieltjes functions because we are interested in the efficiency of numerical application of these approximations. These functions belong to the class of complex-symmetric functions.The PA and NPA to the Stieltjes functions in different regions of the complex plane is also analyzed. It is expected that the appropriate selection of the complex point for the definition of approximant can improve it with respect to the traditional choice of All considered cases are graphically illustrated.Some unique numerical results presented in the paper, which are sufficiently regular should motivate the reader to reflect on them.

Ratios of Entire Functions and Generalized Stieltjes Functions

Monotonicity properties of the ratio $$\begin{aligned} \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, \end{aligned}$$where f is an entire function are investigated. Earlier results for Euler’s gamma function and other entire functions of genus 1 are generalised to entire functions of genus p with negative zeros. Derivatives of order comparable to p of the expression above are related to generalised Stieltjes functions of order \(p+1\). Our results are applied to the Barnes multiple gamma functions. We also show how recent results on the behaviour of Euler’s gamma function on vertical lines can be sharpened and generalised to functions of higher genus. Finally a connection to the so-called Prouhet-Tarry-Escott problem is described.