System Of Orthogonal Polynomials Research Articles

Invariance of the generalized oscillator under a linear transformation of the related system of orthogonal polynomials

We consider two families of polynomials $\mathbb{P}=\polP$ and $\mathbb{Q}=\polQ$\footnote{Here and below we consider only monic polynomials.} orthogonal on the real line with respect to probability measures $\mu$ and $\nu$ respectively. Let $\polQ$ and $\polP$ connected by the linear relations $$ Q_n(x)=P_n(x)+a_1P_{n-1}(x)+...+a_kP_{n-k}(x).$$ Let us denote $\mathfrak{A}_P$ and $\mathfrak{A}_Q$ generalized oscillator algebras associated with the sequences $\mathbb{P}$ and $\mathbb{Q}$. In the case $k=2$ we describe all pairs ($\mathbb{P}$,$\mathbb{Q}$), for which the algebras $\mathfrak{A}_P$ and $\mathfrak{A}_Q$ are equal. In addition, we construct corresponding algebras of generalized oscillators for arbitrary $k\geq1$.

Invariance of the generalized oscillator under a linear transformation of the related system of orthogonal polynomials

Рассматриваются семейства многочленов $\mathbb P=\{P_n(x)\}_{n=0}^\infty$ и $\mathbb Q=\{Q_n(x)\}_{n=0}^\infty$, ортогональных на вещественной прямой относительно вероятностных мер $\mu$ и $\nu$ соответственно. Предполагается, что многочлены $\{Q_n(x)\}_{n=0}^\infty$ и $\{P_n(x)\}_{n=0}^\infty$ связаны линейным соотношением. В случае $k=2$ описаны все пары ($\mathbb P$, $\mathbb Q$), для которых совпадают алгебры $\mathfrak A_P$ и $\mathfrak A_Q$ обобщенных осцилляторов, порождаемые $\{Q_n(x)\}_{n=0}^\infty$ и $\{P_n(x)\}_{n=0}^\infty$. Построены алгебры обобщенных осцилляторов, соответствующие парам ($\mathbb P$, $\mathbb Q$) для произвольного $k\geq 1$.

A review of multivariate orthogonal polynomials

Abstract This paper contains a brief review of orthogonal polynomials in two and several variables. It supplements the Koornwinder survey [40]. Several recently discovered systems of orthogonal polynomials have been treated in this work. We did not provide any proofs of the theorem presented here but references to the original sources are given for the benefit of the interested reader. It is hoped that collecting these scattered results in one place will make them accessible for the user.

On the completely indeterminate case for block Jacobi matrices

Abstract We consider the infinite Jacobi block matrices in the completely indeterminate case, i. e. such that the deficiency indices of the corresponding Jacobi operators are maximal. For such matrices, some criteria of complete indeterminacy are established. These criteria are similar to several known criteria of indeterminacy of the Hamburger moment problem in terms of the corresponding scalar Jacobi matrices and the related systems of orthogonal polynomials.

Computational Analysis of a Nuclear Reactor Cell in the Linear-Anisotropic Approximation by the Generalized Method of First Collisions Probabilities

This work is devoted to the development of a method for solving the Boltzmann kinetic equation for neutron transport. The method is focused on solving an integral equation for neutron transport in a reactor cell with a complex geometry and different boundary conditions. An algorithm is proposed for solving the problem taking account of anisotropic scattering in the linear-anisotropic approximation. The approach is based on expanding the neutron flux in a system of orthogonal two-dimensional polynomials in each uniform zone of a heterogeneous cell. This expansion reduces the system of linear integral equations to a system of linear algebraic equations. The relations required to calculate the coefficients in the equations (six-fold integrals) as well as the computational algorithm are presented. Calculations of cylindrical and cluster cells are presented. The calculations are compared with the surface pseudosource method. It is shown that the results have advantages over the method of first collisions probabilities.

Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

We construct new families of vector orthogonal polynomials that have the property to be eigenfunctions of some differential operator. They are extensions of the Hermite and Laguerre polynomial systems. A third family, whose first member has been found by Y. Ben Cheikh and K. Douak is also constructed. The ideas behind our approach lie in the studies of bispectral operators. We exploit automorphisms of associative algebras which transform elementary vector orthogonal polynomial systems which are eigenfunctions of a differential operator into other systems of this type.

Nonintersecting Brownian motions on the unit circle

We consider an ensemble of $n$ nonintersecting Brownian particles on the unit circle with diffusion parameter $n^{-1/2}$, which are conditioned to begin at the same point and to return to that point after time $T$, but otherwise not to intersect. There is a critical value of $T$ which separates the subcritical case, in which it is vanishingly unlikely that the particles wrap around the circle, and the supercritical case, in which particles may wrap around the circle. In this paper, we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as $n\to\infty$, and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution. We also give a streamlined approach to identifying the Pearcey and tacnode processes in scaling limits. The formula of the tacnode correlation kernel is new and involves a solution to a Lax system for the Painlev\'{e} II equation of size 2 $\times$ 2. The proofs are based on the determinantal structure of the ensemble, asymptotic results for the related system of discrete Gaussian orthogonal polynomials, and a formulation of the correlation kernel in terms of a double contour integral.

Measures for orthogonal polynomials with unbounded recurrence coefficients

Systems of orthogonal polynomials whose recurrence coefficients tend to infinity are considered. A summability condition is imposed on the coefficients and the consequences for the measure of orthogonality are discussed. Also discussed are asymptotics for the polynomials.

A study of resolvent set for a class of band operators with matrix elements

AbstractFor operators generated by a certain class of infinite three-diagonal matrices with matrix elements we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying second order finite-difference equations. This enables us to describe some asymptotic behavior of the corresponding systems of vector orthogonal polynomials on the resolvent set. We also find that the operators generated by infinite Jacobi matrices have the largest resolvent set in this class.

A polynomial expansion to approximate the ultimate ruin probability in the compound Poisson ruin model

A numerical method to approximate ruin probabilities is proposed within the frame of a compound Poisson ruin model. The defective density function associated to the ruin probability is projected in an orthogonal polynomial system. These polynomials are orthogonal with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF). The method is convenient in at least four ways. Firstly, it leads to a simple analytical expression of the ultimate ruin probability. Secondly, the implementation does not require strong computer skills. Thirdly, our approximation method does not necessitate any preliminary discretization step of the claim sizes distribution. Finally, the coefficients of our formula do not depend on initial reserves.

Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

AbstractA 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed thespectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to theq-Painlevé system.

Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

AbstractA 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed thespectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to theq-Painlevé system.

Characterizations of Δ-Volterra lattice: A symmetric orthogonal polynomials interpretation

In this paper we introduce the Δ-Volterra lattice which is interpreted in terms of symmetric orthogonal polynomials. It is shown that the measure of orthogonality associated with these systems of orthogonal polynomials evolves in t like (1+x2)1−tμ(x) where μ is a given positive Borel measure. Moreover, the Δ-Volterra lattice is related to the Δ-Toda lattice from Miura or Bäcklund transformations. The main ingredients are orthogonal polynomials which satisfy an Appell condition with respect to the forward difference operator Δ and the characterization of the point spectrum of a Jacobian operator that satisfies a Δ-Volterra equation (Lax type theorem). We also provide an explicit example of solutions of Δ-Volterra and Δ-Toda lattices, and connect this example with the results presented in the paper.

A quantization of the harmonic analysis on the infinite-dimensional unitary group

The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(∞). That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which replace the nonexisting two-sided regular representation (Olshanski [31]). The required decomposition is governed by certain probability measures on an infinite-dimensional space Ω, which is a dual object to U(∞). A way to describe those measures is to convert them into determinantal point processes on the real line; it turned out that their correlation kernels are computable in explicit form — they admit a closed expression in terms of the Gauss hypergeometric function F12 (Borodin and Olshanski [8]). In the present work we describe a (nonevident) q-discretization of the whole construction. This leads us to a new family of determinantal point processes. We reveal its connection with an exotic finite system of q-discrete orthogonal polynomials — the so-called pseudo big q-Jacobi polynomials. The new point processes live on a double q-lattice and we show that their correlation kernels are expressed through the basic hypergeometric function ϕ12. A crucial novel ingredient of our approach is an extended version G of the Gelfand–Tsetlin graph (the conventional graph describes the Gelfand–Tsetlin branching rule for irreducible representations of unitary groups). We find the q-boundary of G, thus extending previously known results (Gorin [17]).

Asymptotics of determinants of Hankel matrices via non-linear difference equations

E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weight [x(x−α)(x−β)]−12, x∈[0,α], 0<α<β. A related system was studied by C. J. Rees in 1945, associated with the weight [(1−x2)(1−k2x2)]−12, x∈[−1,1], k2∈(0,1). These are also known as elliptic orthogonal polynomials, since the moments of the weights may be expressed in terms of elliptic integrals. Such orthogonal polynomials are of great interest because the corresponding Hankel determinant, depending on a parameter k2, where 0<k2<1 is the τ function of a particular Painlevé VI, the special cases of which are related to enumerative problems arising from string theory. We show that the recurrence coefficients, denoted by βn(k2),n=1,2,…; and p1(n,k2), the coefficients of xn−2 of the monic polynomials orthogonal with respect to a generalized version of the weight studied by Rees, (1−x2)α(1−k2x2)β,x∈[−1,1],α>−1,β∈R, satisfy second order non-linear difference equations. The large n expansion based on the difference equations when combined with known asymptotics of the leading terms of the associated Hankel determinant yields a complete asymptotic expansion of the Hankel determinant. The Painlevé equation is also discussed as well as the generalization of the linear second order differential equation found by Rees.

Six-vertex model with partial domain wall boundary conditions: Ferroelectric phase

We obtain an asymptotic formula for the partition function of the six-vertex model with partial domain wall boundary conditions in the ferroelectric phase region. The proof is based on a formula for the partition function involving the determinant of a matrix of mixed Vandermonde/Hankel type. This determinant can be expressed in terms of a system of discrete orthogonal polynomials, which can then be evaluated asymptotically by comparison with the Meixner polynomials.

Inequalities for some systems of orthogonal polynomials using reproducing kernel functions

In this paper we are engaged with classical orthogonal polynomials. We deduce relations for the reproducing kernels for these polynomials without derivations. By the using this form of reproducing kernels of classical orthogonal polynomials we deduce some inequalities for polynomials with special weight function. This enabled us to derive some inequalities for orthonormal polynomial with special weight function. It allows us to deduce general statement for every orthonormal polynomial with a nonnegative weight function on a finite interval of orthogonality. Mathematics Subject Classification: 33C45, 33C47

On some special polynomials appearing in orthogonal systems on the unit circle

Abstract A classical theorem due to Eneström and Kakeya gives some bounds for the moduli of the zeros of polynomials having a monotone sequence of non-negative (real) coefficients. Nowadays, one can find several modifications and generalizations of this result in the literature. The main subject of the paper is a study of links of coefficients which occur in some of these general results with a view to the recurrence relations fulfilled by systems of orthogonal polynomials on the unit circle. In particular, we discuss the question, if the relevant links are consistent with the recurrence relations or not. This leads to some new insight into the analyzed classes of polynomials.

A Matrix Recurrence for Systems of Clifford Algebra-Valued Orthogonal Polynomials

Recently, the authors developed a matrix approach to multivariate polynomial sequences by using methods of Hypercomplex Function Theory (Matrix representations of a basic polynomial sequence in arbitrary dimension. Comput. Methods Funct. Theory, 12 (2012), no. 2, 371-391). This paper deals with an extension of that approach to a recurrence relation for the construction of a complete system of orthogonal Clifford-algebra valued polynomials of arbitrary degree. At the same time the matrix approach sheds new light on results about systems of Clifford algebra-valued orthogonal polynomials obtained by Gürlebeck, Bock, Lávička, Delanghe et al. during the last five years. In fact, it allows to prove directly some intrinsic properties of the building blocks essential in the construction process, but not studied so far.

Complexity analysis of hypergeometric orthogonal polynomials

The complexity measures of the Crámer–Rao, Fisher–Shannon and LMC (López-Ruiz, Mancini and Calvet) types of the Rakhmanov probability density ρn(x)=ω(x)pn2(x) of the polynomials pn(x) orthogonal with respect to the weight function ω(x), x∈(a,b), are used to quantify various two-fold facets of the spreading of the Hermite, Laguerre and Jacobi systems all over their corresponding orthogonality intervals in both analytical and computational ways. Their explicit (Crámer–Rao) and asymptotical (Fisher–Shannon, LMC) values are given for the three systems of orthogonal polynomials. Then, these complexity-type mathematical quantities are numerically examined in terms of the polynomial’s degree n and the parameters which characterize the weight function. Finally, several open problems about the generalized hypergeometric functions of Lauricella and Srivastava-Daoust types, as well as on the asymptotics of weighted Lq-norms of Laguerre and Jacobi polynomials are pointed out.