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1.
In the present article, we investigate the properties of bivariate Fibonacci polynomials of order k in terms of the generating functions. For k and ℓ (1 ≤ ℓ ≤ k − 1), the relationship between the bivariate Fibonacci polynomials of order k and the bivariate Fibonacci polynomials of order ℓ is elucidated. Lucas polynomials of order k are considered. We also reveal the relationship between Lucas polynomials of order k and Lucas polynomials of order ℓ. The present work extends several properties of Fibonacci and Lucas polynomials of order k, which will lead us a new type of geneses of these polynomials. We point out that Fibonacci and Lucas polynomials of order
k are closely related to distributions of order k and show that the distributions possess properties analogous to the bivariate Fibonacci and Lucas polynomials of order k. 相似文献
2.
In this paper, we consider the zero distributions of q-shift difference polynomials of meromorphic functions with zero order, and obtain two theorems that extend the classical
Hayman results on the zeros of differential polynomials to q-shift difference polynomials. We also investigate the uniqueness problem of q-shift difference polynomials that share a common value. 相似文献
3.
We study the degree distribution of the greatest common divisor of two or more random polynomials over a finite field ??q. We provide estimates for several parameters like number of distinct common irreducible factors, number of irreducible factors counting repetitions, and total degree of the gcd of two or more polynomials. We show that the limiting distribution of a random variable counting the total degree of the gcd is geometric and that the distributions of random variables counting the number of common factors (with and without repetitions) are very close to Poisson distributions when q is large. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006 相似文献
4.
Summary The class of discrete distributions of orderk is defined as the class of the generalized discrete distributions with generalizer a discrete distribution truncated at zero
and from the right away fromk+1. The probability function and factorial moments of these distributions are expressed in terms of the (right) truncated
Bell (partition) polynomials and several special cases are briefly examined. Finally a Poisson process of orderk, leading in particular to the Poisson distribution of orderk, is discussed. 相似文献
5.
Kenneth S. Berenhaut John V. Baxley Robert G. Lyday 《Statistics & probability letters》2011,81(12):1940-1944
In this note, we consider a question of Móri regarding estimating the deviation of the kth terms of two discrete probability distributions in terms of the supremum distance between their generating functions over the interval [0,1]. An optimal bound for distributions on finite support is obtained. Properties of Chebyshev polynomials are employed. 相似文献
6.
A. W. Davis 《Annals of the Institute of Statistical Mathematics》1982,34(1):517-521
Summary Roy and Gnanadesikan [5] showed that inference for a general multivariate variance components model may be carried out using
the standard multivariateF distribution under certain condtions. It is shown in this note that the theory of zonal polynomials, and their extension
by the author to invariant polynomials in two matrix arguments, provide a concise approach to the derivation of these conditions.
Relevant distributions are also derived for the general case.
CSIRO 相似文献
7.
V. Papathanasiou 《Annals of the Institute of Statistical Mathematics》1995,47(1):171-176
A new derivation of the classical orthogonal polynomials is given by using thew-function which appears in the variance bounds and some properties of the Pearson system of distributions. Also a characterization of the Pearson system of distributions through some conditional moments is obtained by using a result obtained by Johnson (1993) concerning this family. 相似文献
8.
We define multivariate Meixner classes of invariant distributions of random matrices as those whose generating functions for the associated orthogonal polynomials are of certain special integral or summation forms, generalizing the univariate Meixner classes of distributions which were first characterized by Meixner [21]. Characterization theorems and properties of these multivariate Meixner classes are established. The zonal polynomials, the extended invariant polynomials with matrix arguments, and their related results in multivariate distribution theory are utilized in the discussion. 相似文献
9.
D. G. Marx 《Annals of the Institute of Statistical Mathematics》1983,35(1):347-353
Summary Crowther [2] studied the distribution of a quadratic form in a matrix normal variate. This, in some sense, is extended by
De Waal [4]. They represented the density function of this quadratic form in terms of generalized Hayakawa polynomials. Application
of some specific results of these authors facilitates the derivation of distributions of quadratic forms of the matric-t variate. Attention is also given to the distributions of the characteristic roots and the trace of this quadratic matrix.
Special cases are considered and some useful integrals are formulated.
Financially supported by the CSIR and the University of the Orange Free State 相似文献
10.
Jacobi polynomials were introduced by Ozeki in analogy with Jacobi forms of lattices. They are useful to compute coset weight
enumerators, and weight enumerators of children. We determine them in most interesting cases in length at most 32, and in
some cases in length 72. We use them to construct group divisible designs, packing designs, covering designs, and (t,r)-designs in the sense of Calderbank-Delsarte. A major tool is invariant theory of finite groups, in particular simultaneous
invariants in the sense of Schur, polarization, and bivariate Molien series. A combinatorial interpretation of the Aronhold
polarization operator is given. New rank parameters for spaces of coset weight distributions and Jacobi polynomials are introduced
and studied here. 相似文献
11.
Elchanan Mossel 《Geometric And Functional Analysis》2010,19(6):1713-1756
In this paper we derive tight bounds on the expected value of products of low influence functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number
of correlated probability spaces, on a generalization of an invariance principle recently obtained with O’Donnell and Oleszkiewicz
for multilinear polynomials with low influences and bounded degree and on properties of multi-dimensional Gaussian distributions. 相似文献
12.
Prof. Dr. Peter Lesky 《Monatshefte für Mathematik》1984,98(4):277-293
Using the Pearson difference equation for the discrete classical orthogonal polynomials the difference equations and the Rodrigues formulas are obtained. The resulting weight functions prove to be the probability functions of the most important discrete probability distributions: Pólya distribution from the Hahn and Krawtchouk polynomials, negative binomial distribution from the Meixner polynomials, Poisson distribution from the Charlier polynomials. 相似文献
13.
Satoshi Kuriki Yasuhide Numata 《Annals of the Institute of Statistical Mathematics》2010,62(4):645-672
We provide formulas for the moments of the real and complex noncentral Wishart distributions of general degrees. The obtained
formulas for the real and complex cases are described in terms of the undirected and directed graphs, respectively. By considering
degenerate cases, we give explicit formulas for the moments of bivariate chi-square distributions and 2 × 2 Wishart distributions
by enumerating the graphs. Noting that the Laguerre polynomials can be considered to be moments of a noncentral chi-square
distributions formally, we demonstrate a combinatorial interpretation of the coefficients of the Laguerre polynomials. 相似文献
14.
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrixrnew rows and columns, so that the original Jacobi matrix is shifted downward. Thernew rows and columns contain 2rnew parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials, and the 2rnew parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the 1?orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials. 相似文献
15.
Michael Voit 《Studies in Applied Mathematics》2023,151(4):1230-1281
One-dimensional interacting particle models of Calogero–Moser–Sutherland type with N particles can be regarded as diffusion processes on suitable subsets of like Weyl chambers and alcoves with second-order differential operators as generators of the transition semigroups, where these operators are singular on the boundaries of the state spaces. The most relevant examples are multivariate Bessel processes and Heckman–Opdam processes in a compact and noncompact setting where in all cases, these processes are related to special functions associated with root systems. More precisely, the transition probabilities can be described with the aid of multivariate Bessel functions, Jack and Heckman–Opdam Jacobi polynomials, and Heckman–Opdam hypergeometric functions, respectively. These models, in particular, form dynamic eigenvalue evolutions of the classical random matrix models like β-Hermite, β- Laguerre, and β-Jacobi, that is, MANOVA, ensembles. In particular, Dyson's Brownian motions and multivariate Jacobi processes are included. In all cases, the processes depend on so-called coupling parameters. We review several freezing limit theorems for these diffusions where, for fixed N, one or several of the coupling parameters tend to ∞. In many cases, the limits will be N-dimensional normal distributions and, in the process case, Gauss processes. However, in some cases, normal distributions on half spaces and distributions related to some other ensembles appear as limits. In all cases, the limits are connected with the zeros of the classical one-dimensional orthogonal polynomials of order N. 相似文献
16.
A. Draux 《Integral Transforms and Special Functions》2016,27(9):747-765
The sequences of quasi-orthogonal polynomials of order r are defined for non-quasi-definite moment functionals. Properties concerning the existence of such sequences, and relations between a quasi-orthogonal polynomial of order r and a set of orthogonal polynomials are proved. Two determinantal expressions of quasi-orthogonal polynomials of order r are given. At last it is proved that three consecutive polynomials of a sequence of quasi-orthogonal polynomials of order r satisfy a three term recurrence relation. 相似文献
17.
Bayes-empiric Bayes estimation of the parameter of certain one parameter discrete exponential families based on orthogonal polynomials on an interval (a, b) is introduced. The resulting estimator is shown to be asymptotically optimal. The application of this method to three special distributions, the binomial, Poisson and negative binomial, is discussed.The first author was supported by NSF grant DCR-8504620. 相似文献
18.
J. Liesen 《Constructive Approximation》2001,17(2):267-274
Faber polynomials corresponding to rational exterior mapping functions of degree (m, m − 1) are studied. It is shown that these polynomials always satisfy an (m + 1)-term recurrence. For the special case m = 2, it is shown that the Faber polynomials can be expressed in terms of the classical Chebyshev polynomials of the first
kind. In this case, explicit formulas for the Faber polynomials are derived. 相似文献
19.
Yu. I. Kuznetsov 《Siberian Mathematical Journal》2001,42(6):1093-1101
We consider the classical problem of transforming an orthogonality weight of polynomials by means of the space R
n
. We describe systems of polynomials called pseudo-orthogonal on a finite set of n points. Like orthogonal polynomials, the polynomials of these systems are connected by three-term relations with tridiagonal matrix which is nondecomposable but does not enjoy the Jacobi property. Nevertheless these polynomials possess real roots of multiplicity one; moreover, almost all roots of two neighboring polynomials separate one another. The pseudo-orthogonality weights are partly negative. Another result is the analysis of relations between matrices of two different orthogonal systems which enables us to give explicit conditions for existence of pseudo-orthogonal polynomials. 相似文献
20.
In this note are deduced two characterizations of normal distributions in the class NH of probability distributions on a plane whose densities admit diagonal expansions in series of Hermite polynomials and the marginal distributions are standardized and normal.Translated from Matematicheskie Zametkl, Vol. 20, No. 1, pp. 139–146, July, 1976. 相似文献