Hahn Polynomials Research Articles

Hahn polynomials for hypergeometric distribution

Orthogonal polynomials for the multivariate hypergeometric distribution are defined on lattices in polyhedral domains in Rd. Their structures are studied through a detailed analysis of classical Hahn polynomials with negative integer parameters. Factorization of the Hahn polynomials is explored and used to explain the relationship between the index set of orthogonal polynomials and the lattice set in the polyhedral domain. In the multivariate case, these constructions lead to nontrivial families of hypergeometric polynomials vanishing on lattice polyhedra. The generating functions and bispectral properties of the orthogonal polynomials are also discussed.

An urn model for the Jacobi-Piñeiro polynomials

The list of physically motivated urn models that can be solved in terms of classical orthogonal polynomials is very small. It includes a model proposed by D. Bernoulli and further analyzed by S. Laplace and a model proposed by P. and T. Ehrenfest and eventually connected with the Krawtchouk and Hahn polynomials. This connection was reversed recently in the case of the Jacobi polynomials where a rather contrived, and later a simpler urn model was proposed. Here we consider an urn model going with the Jacobi-Piñeiro multiple orthogonal polynomials. These polynomials have recently been put forth in connection with a stochastic matrix.

Tight relative t-designs on two shells in hypercubes, and Hahn and Hermite polynomials

Relative $t$-designs in the $n$-dimensional hypercube $\mathcal{Q}_n$ are equivalent to weighted regular $t$-wise balanced designs, which generalize combinatorial $t$-$(n,k,\lambda)$ designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean $t$-designs on two concentric spheres, in this paper we discuss tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells. We show under a mild condition that such a relative $t$-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells are rare for large $t$.

Optimal reconstruction and compression of signals and images by Hahn moments and artificial bee Colony (ABC) algorithm.

In this paper, we present an efficient and optimal method for optimization of Hahn parameters a and b using the Artificial Bee Colony algorithm (ABC) in order to improve the quality of reconstruction and the compression of bio-signals and 2D / 3D color images of large sizes. The proposed methods are essentially based on two concepts: the development of a recursive calculation of the initial terms of Hahn polynomials in order to avoid the problems of instability of polynomial values and the use of ABC algorithm to optimize the values of the parameters a and b of the discrete orthogonal Hahn polynomials (HPs) during the reconstruction and the compression of bio-signals and 2D / 3D color images. The simulation results performed on bio-signals and on large size 2D /3D color images clearly show the efficiency and superiority of the proposed methods over conventional methods in terms of reconstruction of signals and images.

Moments of orthogonal polynomials and exponential generating functions

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ( $$q=1$$ ) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont–Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally, we show that one of our methods can be applied to deal with the moments of Askey–Wilson polynomials.

Robust 2D and 3D images zero - watermarking using dual Hahn moment invariants and Sine Cosine Algorithm.

In this paper, we initially provide significant improvements on the computational aspects of dual Hahn moment invariants (DHMIs) in both 2D and 3D domains. These improvements ensure the numerical stability of DHMIs for large-size images. Then, we propose an efficient method for optimizing the local parameters of dual Hahn polynomials (DHPs) when computing DHMIs using the Sine-Cosine Algorithm (SCA). DHMIs optimized via SCA are used to propose new and robust zero-watermarking scheme applied to both 2D and 3D images. On one hand, the simulation results confirm the efficiency of the proposed computation of 2D and 3D DHMIs regarding the numerical stability and invariability. Indeed, the proposed computation method of 2D DHMIs allows to reach a relative error (RE) of the order ≈10−10 for images of size 1024 × 1024 with an execution time improvement ratio exceeds 70% (ETIR ≥ 70%), which validates the efficiently of the proposed computation method. On the other hand, the simulation and comparison outcomes clearly demonstrate the robustness of the proposed zero-watermarking scheme against various geometric attacks (translation, rotation, scaling and combined transformations), as well as against other common 2D and 3D image processing attacks (compression, filtering, noise addition, etc.).Graphical abstract

On computational aspects of high-order dual Hahn moments

In this paper, we present two new algorithms for the fast and stable computation of high-order discrete orthogonal dual Hahn polynomials (DHPs). These algorithms are essentially based on the proposed computation method of the initial values of DHPs following the order n and the variable s. For both algorithms, a single stable value is computed, fully independent of the gamma function that is the source of the numerical overflow, and then the rest of DHPs values are computed recursively via the proposed recurrence scheme. By analyzing the DHPs matrix, we propose a new method, which allows ensuring the numerical stability of high-order DHPs and dual Hahn moments (DHMs) until the last order. This method is based on the use of appropriate stability conditions. The results of simulations and comparisons carried out show on one hand that the second algorithm with the stability condition allows to compute DHPs up to the order n = 17,603 without propagation of numerical error. On the other hand, the performance of analyzing large-size signals and images by high-order DHMs computed by the proposed method significantly exceeds the existing methods in terms of numerical stability, accuracy of reconstruction and in terms of maximum size of the analyzed signals and images. After the acceptance of this paper, the proposed algorithms for high-order DHPs computation will be made publically available at https://github.com/AchrafDaoui/On-Computational-Aspects-of-High-Order-Dual-Hahn-Moments.

Fast Computation of Hahn Polynomials for High Order Moments

Discrete Hahn polynomials (DHPs) and their moments are considered to be one of the efficient orthogonal moments and they are applied in various scientific areas such as image processing and feature extraction. Commonly, DHPs are used as object representation; however, they suffer from the problem of numerical instability when the moment order becomes large. In this paper, an operative method to compute the Hahn orthogonal basis is proposed and applied to high orders. This paper developed a new mathematical model for computing the initial value of the DHP and for different values of DHP parameters (<inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula>). In addition, the proposed method is composed of two recurrence algorithms with an adaptive threshold to stabilize the generation of the DHP coefficients. It is compared with state-of-the-art algorithms in terms of computational cost and the maximum size that can be correctly generated. The experimental results show that the proposed algorithm performs better in both parameters for wide ranges of parameter values <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula>, and polynomial sizes.

New examples of Krall–Meixner and Krall–Hahn polynomials, with applications to the construction of exceptional Meixner and Laguerre polynomials

We construct new examples of Krall discrete orthogonal polynomials, i.e., orthogonal polynomials with respect to a measure which are also eigenfunctions of a higher order difference operator. The new examples include the orthogonal polynomials with respect to the measures obtained from the Meixner measure ρa,c and Hahn measure ρa,b,N by removing a finite number of their mass points when the parameter c of the Meixner measure and a or b of the Hahn measure are positive integers. From the new Krall–Meixner families we construct new families of exceptional Meixner and Laguerre polynomials.

On the continuous dual Hahn process

We extend the continuous dual Hahn process (Tt) of Corwin and Knizel from a finite time interval to the entire real line by taking a limit of a closely related Markov process (Tt). We also characterize processes (Tt) by conditional means and variances under bidirectional conditioning, and we prove that continuous dual Hahn polynomials are orthogonal martingale polynomials for both processes.

An efficient approach based on Legendre–Gauss–Lobatto quadrature and discrete shifted Hahn polynomials for solving Caputo–Fabrizio fractional Volterra partial integro-differential equations

In the current study, we provide a novel technique based on discrete shifted Hahn polynomials and Legendre–Gauss–Lobatto quadrature method for solving Caputo–Fabrizio fractional Volterra partial integro-differential equations (CFF-VPIDEs). The process of numerical algorithm contains the modified operational matrices (MOMs) and complement vectors (CVs), which directly influence the accuracy of the approach. Also, we expand the Volterra integral part of the equation with the help of the Legendre–Gauss–Lobatto quadrature method. The composition of the novel operational matrices with the Legendre–Gauss–Lobatto quadrature rule creates high precision and efficient method. Further, we present the error analysis of our scheme. Finally, to confirm the accuracy of theoretical results, we examine several examples.

Optimal data processing in the ballistic laser gravimeter under the effect of correlated interference

Abstract&#x0D; In order to develop optimal data processing algorithms in ballistic laser gravimeters under the effect of correlated interference, the method of generalized least squares is applied. In this case, to describe the interference, a mathematical model of the autoregression process is used, for which the inverse correlation matrix has a band type and is expressed through the values of the autoregression coefficients. To convert the “path-time” data from the output of the coincidence circuit of ballistic laser gravimeters to a process uniform in time, their local quadratic interpolation is used.&#x0D; Algorithms for data processing in a ballistic gravimeter, developed on the basis of a method of weighted least squares using orthogonal Hahn polynomials, are considered. To implement a symmetric measurement method, the symmetric Hahn polynomials, characterized by one parameter, are used.&#x0D; The method of mathematical modelling is used to study the gain in the accuracy of measuring the gravitational acceleration by the synthesized algorithms in comparison with the algorithm based on the method of least squares. It is shown that auto seismic interference in ballistic laser gravimeters with a symmetric measurement method can be significantly reduced by using a mathematical model of the second-order autoregressive process in the method of generalized least squares. A comparative analysis of the characteristics of the algorithms developed using the method of generalized least squares, the method of weighted least squares and the method of ordinary least squares is carried out.

Exceptional Hahn and Jacobi polynomials with an arbitrary number of continuous parameters

AbstractWe construct new examples of exceptional Hahn and Jacobi polynomials. Exceptional polynomials are orthogonal polynomials with respect to a measure which are also eigenfunctions of a second‐order difference or differential operator. In mathematical physics, they allow the explicit computation of bound states of rational extensions of classical quantum‐mechanical potentials. The most apparent difference between classical or classical discrete orthogonal polynomials and their exceptional counterparts is that the exceptional families have gaps in their degrees, in the sense that not all degrees are present in the sequence of polynomials. The new examples have the novelty that they depend on an arbitrary number of continuous parameters.

Orthogonal polynomials and the deformed Jordan plane

We consider the unital associative algebra A with two generators X, Z obeying the defining relation [Z,X]=Z2+Δ. We construct irreducible tridiagonal representations of A. Depending on the value of the parameter Δ, these representations are associated to the Jacobi matrices of the para-Krawtchouk, continuous Hahn, Hahn or Jacobi polynomials.

Hahn polynomials and the Burnside process

We study a natural Markov chain on \(\{0,1,\ldots ,n\}\) with eigenvectors the Hahn polynomials. This explicit diagonalization makes it possible to get sharp rates of convergence to stationarity. The process, the Burnside process, is a special case of the celebrated ‘Swendsen–Wang’ or ‘data augmentation’ algorithm. The description involves the beta-binomial distribution and Mallows model on permutations. It introduces a useful generalization of the Burnside process.

Fast and stable computation of higher-order Hahn polynomials and Hahn moment invariants for signal and image analysis.

This article presents, on the one hand, new algorithms for the fast and stable computation of discrete orthogonal Hahn polynomials of high order (HPs) based on the elimination of all gamma and factorial functions that cause the numerical fluctuations of HPs, and based on the use of appropriate stability conditions. On the other hand, a new method for the fast and numerically stable computation of Hahn moment invariants (HMIs) is also proposed. This method is mainly based on the use of new recursive relations of HPs and of matrix multiplications when calculating HMIs. To validate the efficiency of the algorithms proposed for the calculation of HPs, several signals and large images (≥4000 × 4000) are reconstructed by Hahn moments (HMs) up to the last order with a reconstruction error tending towards zero (MSE ≃ 10−10). The efficiency of the proposed method for calculating HMIs is demonstrated on large medical images (2048 × 2048) with a very low relative error (RE ≃ 10−10). Finally, comparisons with some recent work in the literature are provided.

Relations between moments for the Jacobi and Cauchy random matrix ensembles

We outline a relation between the densities for the β-ensembles with respect to the Jacobi weight (1 − x)a(1 + x)b supported on the interval (−1, 1) and the Cauchy weight (1−ix)η(1+ix)η̄ by appropriate analytic continuation. This has the consequence of implying that the latter density satisfies a linear differential equation of degree three for β = 2 and of degree five for β = 1 and 4, analogs of which are already known for the Jacobi weight xa(1 − x)b supported on (0, 1). We concentrate on the case a = b [Jacobi weight on (−1, 1)] and η real (Cauchy weight) since the density is then an even function and the differential equations simplify. From the differential equations, recurrences can be obtained for the moments of the Jacobi weight supported on (−1, 1) and/or the moments of the Cauchy weight. Particular attention is paid to the case β = 2 and the Jacobi weight on (−1, 1) in the symmetric case a = b, which in keeping with a recent result obtained by Assiotis et al. (“Moments of generalised Cauchy random matrices and continuous-Hahn polynomials,” Nonlinearity (to be published), arXiv:2009.04752) for the β = 2 case of the symmetric Cauchy weight (parameter η real), allows for an explicit solution of the recurrence in terms of particular continuous Hahn polynomials. Also for the symmetric Cauchy weight with η = −β(N − 1)/2 − 1 − α, after appropriately scaling α proportional to N, we use differential equations to compute terms in the 1/N2 (1/N) expansion of the resolvent for β = 2 (β = 1, 4).

Orthonormal shifted discrete Hahn polynomials for a new category of nonlinear variable‐order fractional 2D optimal control problems

AbstractThe present work provides a formulation for the orthonormal shifted discrete Hahn polynomials on an arbitrary domain. We also extract some useful matrix relationshipsrelating to these discrete polynomials. These polynomials and the obtained relations are employed with the Lagrange multipliers technique to design a numerical method for solving a class of variable‐order (VO) fractional 2D optimization problems subject to Robin boundary value conditions. The proposed method converts the VO fractional system into a system of algebraic equations which readily can be solved. Some numerical examples are studied to validate and illustrate the accuracy of the introduced technique.

A class of Sobolev orthogonal polynomials on the unit circle and associated continuous dual Hahn polynomials: Bounds, asymptotics and zeros

This paper deals with orthogonal polynomials and associated connection coefficients with respect to a class of Sobolev inner products on the unit circle. Under certain conditions on the parameters in the inner product it is shown that the connection coefficients are related to a subfamily of the continuous dual Hahn polynomials. Properties regarding bounds and asymptotics are also established with respect to these parameters. Criteria for knowing when the zeros of the (Sobolev) orthogonal polynomials and also the zeros of their derivatives stay within the unit disk have also been addressed. By numerical experiments some further information on the parameters is also found so that the zeros remain within the unit disk.

The family of confluent Virasoro fusion kernels and a non-polynomial $q$-Askey scheme

We study the recently introduced family of confluent Virasoro fusion kernels $\mathcal{C}_k(b,\boldsymbol{\theta},\sigma_s,\nu)$. We study their eigenfunction properties and show that they can be viewed as non-polynomial generalizations of both the continuous dual $q$-Hahn and the big $q$-Jacobi polynomials. More precisely, we prove that: (i) $\mathcal{C}_k$ is a joint eigenfunction of four different difference operators for any positive integer $k$, (ii) $\mathcal{C}_k$ degenerates to the continuous dual $q$-Hahn polynomials when $\nu$ is suitably discretized, and (iii) $\mathcal{C}_k$ degenerates to the big $q$-Jacobi polynomials when $\sigma_s$ is suitably discretized. These observations lead us to propose the existence of a non-polynomial generalization of the $q$-Askey scheme. The top member of this non-polynomial scheme is the Virasoro fusion kernel (or, equivalently, Ruijsenaars' hypergeometric function), and its first confluence is given by the $\mathcal{C}_k$.