On the orthogonality and convolution orthogonality via the Kontorovich–Lebedev transform

A family of multivariate orthogonal polynomials generalizing the standard (univariate) Charlier polynomials is shown to arise in the matrix elements of the unitary representation of the Euclidean group E(d) on oscillator states. These polynomials in d discrete variables are orthogonal on the product of d Poisson distributions. The accent is put on the d = 2 case and the group theoretical setting is used to obtain the main properties of the polynomials: orthogonality and recurrence relations, difference equation, raising/lowering relations, generating function, hypergeometric and integral representations and explicit expression in terms of standard Charlier and Krawtchouk polynomials. The approach is seen to extend straightforwardly to an arbitrary number of variables. The contraction of SO(3) to E(2) is used to show that the bivariate Charlier polynomials correspond to a limit of the bivariate Krawtchouk polynomials.

We consider the eigenvalue problem associated with the Dunkl-type differential operator (in which the reflection operator R is involved)in the context of supersymmetric quantum mechanical models. By solving this eigenvalue problem with the help of known exactly solvable potentials, we construct several classes of functions satisfying certain orthogonality relations. We call them the Dunkl-supersymmetric (Dunkl-SUSY) orthogonal functions. These functions can be expressed in terms of the classical orthogonal polynomials. The key feature of these functions is that they appear by pairs, i.e. Qn(x) and Qn(−x) are both the eigenfunctions of . A general formulation of the Dunkl-SUSY orthogonal polynomials is also presented.

Orthogonal polynomials suggested by a queueing model

On the frequency mapping of modal parameters identification

The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials whose properties give the structure and dynamics of the corresponding physical system. For a certain range of parameters, one of these polynomials has a mix of continuous and discrete spectra making it suitable for describing physical systems with both scattering and bound states. In this work, we define these polynomials by their recursion relations and highlight some of their properties using numerical means. Due to the prime significance of these polynomials in physics, we hope that our short expose will encourage experts in the field of orthogonal polynomials to study them and derive their properties (weight functions, generating functions, asymptotics, orthogonality relations, zeros, etc.) analytically.

Classical orthogonal polynomials, generalized and shifted orthogonal polynomials serve as an excellent tool for modelling plants and processes in an approximative way. Moreover, solutions via orthogonal polynomials give a straightforward algorithm in reducing various dynamic problems to the solution of a set of algebraic linear equations. The integral of a function with respect to time is replaced by premultiplying the vector of the decomposed signal by an operational matrix. Furthermore, the product of functions and even timevarying coefficients and nonlinear relations are reduced to algebraic manipulations. Among these problems are system analysis, variational calculus, optimal control, identification and model reduction. Orthogonal functions have been applied in control measurement and correlation techniques for a long period, see e.g. Kitamori, T., 1960; Douce, J.L., and Roberts, P.D., 1964. The classical Fourier series expansion, truncated to a certain amount of terms, can also be applied to solve various dynamic problems (Paraskevopoulos, P.N., et al. 1985; Cheok, K.C., et al. 1989).

We show that the Stirling numbers of the first and second kind can be represented in terms of moments of appropriate random variables. This allows us to obtain, in a unified way, a variety of old and new explicit expressions and integral representations of such numbers.

The weight functions, generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials

We consider an interpolation problem of Nevanlinna–Pick type for matrix‐valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non‐degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix‐valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix‐valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In the paper a comparison is made between two approximate methods of system analysis based on the use of the operational matrices of integration and differentiation. Both methods applied to model problems involving linear time-invariant differential equations yielded results of equal order of accuracy for the same number of orthogonal polynomial terms. However, the operational matrix of differentiation method is not limited to initial-value problems, but may easily be applied to boundary-value problems. In addition, due to the exact definition of this matrix, transformations from an orthogonal polynomial base to another may easily be performed with the use of the operational matrix of polynomial series transformation. The operational matrix of differentiation approach may be easily applied to the analysis of other types of systems such as time-varying and distributed parameter systems, and can be easily extended for other types of orthogonal differentiable base functions such as the Fourier trigonometric functions and the Bessel functions.

Some sets of orthogonal polynomial kernel functions

Approximation of structure functions (SFs) using orthogonal polynomials in most cases, leads to an incorrect determination of the spectral power density for the obtained orthogonal model. A method for constructing orthogonal models of SFs is proposed based on preliminary shifting of the analyzed SF and using orthogonal functions as the bases. In this method, the expansion coefficients are determined using a numerical-analytical approach, which allows one to determine orthogonal models of SFs with a small approximation error and calculate spectral power density with greater accuracy. The implementation of the algorithms for constructing orthogonal models of SFs in the Laguerre and Legendre bases is considered.

A method for analyzing and evaluating form errors of cylindrical parts is investigated. The profile of a cylindrical test piece is measured at several sections along the axis under the same setting on the rotating table of a roundness measuring machine which has an accurate vertical feed of the detector. An orthogonal function consisting of Fourier series and orthogonal polynomials is fitted to the observed data in cylindrical coordinates sampled at every 7.5 degrees for each section of the test piece. Geometrical meanings of individual terms of the orthogonal function are disclosed. By dissolving the total sum of squares of the observations into parts corresponding to individual terms and testing their significances against the residual variance, the form erorrs of the test piece as well as the errors in measurement are assessed independently of the setting conditions. The method is applied for measuring form errors in a few plug gauges and cylindrical machine parts.

After a short review of the quantum mechanics canonically associated with a classical real valued random variable with all moments, we begin to study the quantum mechanics canonically associated to the standard semi-circle random variable [Formula: see text], characterized by the fact that its probability distribution is the semi-circle law [Formula: see text] on [Formula: see text]. We prove that, in the identification of [Formula: see text] with the [Formula: see text]-mode interacting Fock space [Formula: see text], defined by the orthogonal polynomial gradation of [Formula: see text], [Formula: see text] is mapped into position operator and its canonically associated momentum operator [Formula: see text] into [Formula: see text] times the [Formula: see text]-Hilbert transform [Formula: see text] on [Formula: see text]. In the first part of the present paper, after briefly describing the simpler case of the [Formula: see text]-harmonic oscillator, we find an explicit expression for the action, on the [Formula: see text]-orthogonal polynomials, of the semi-circle analogue of the translation group [Formula: see text] and of the semi-circle analogue of the free evolution [Formula: see text], respectively, in terms of Bessel functions of the first kind and of confluent hyper-geometric series. These results require the solution of the inverse normal order problem on the quantum algebra canonically associated to the classical semi-circle random variable and are derived in the second part of the present paper. Since the problem to determine, with purely analytic techniques, the explicit form of the action of [Formula: see text] and [Formula: see text] on the [Formula: see text]-orthogonal polynomials is difficult, the above mentioned results show the power of the combination of these techniques with those developed within the algebraic approach to the theory of orthogonal polynomials.

The probability generating function (pgf) B ⁢ ( s ) {B(s)} of the limiting conditional law (LCL) of a subcritical Markov branching process ( Z t : t ≥ 0 ) {(Z_{t}:t\geq 0)} (MBP) has a certain integral representation and it satisfies B ⁢ ( 0 ) = 0 {B(0)=0} and B ′ ⁢ ( 0 ) > 0 {B^{\prime}(0)>0} . The general problem posed here is the inverse one: If a given pgf B satisfies these two conditions, is it related in this way to some MBP? We obtain some necessary conditions for this to be possible and illustrate the issues with simple examples and counterexamples. The particular case of the Borel law is shown to be the LCL of a family of MBPs and that the probabilities P 1 ( Z t = j ) {P_{1}(Z_{t}=j)} have simple explicit algebraic expressions. Exact conditions are found under which a shifted negative-binomial law can be a LCL. Finally, implications are explored for the offspring law arising from infinite divisibility of the correponding LCL.