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In this paper, we propose a new fast way to compute both the image Charlier moments and its inverses using Clenshaw’s recurrence formula. Firstly, we present recursive polynomials of Charlier with respect to the order n and with respect to the variable x and then we define Clenshaw’s recurrence formula to improve the consuming time of the proposed algorithm. So, to show the robustness of the proposed method, a comparative study with the classical method is carried out. In fact, the results of the simulations carried out on binary and gray-scale images show the effectiveness of the proposed method in terms of the calculation time of Charlier moments and in terms of image reconstruction capacity with respect to Krawtchouk moments.

The main propose of this article is to investigate and modify Hermite type polynomials, Milne-Thomson type polynomials and Poisson–Charlier type polynomials by using generating functions and their functional equations. By using functional equations of the generating functions for these polynomials, we not only derive some identities and relations including the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Poisson–Charlier polynomials, the Milne-Thomson polynomials and the Hermite polynomials, but also study some fundamental properties of these functions and polynomials. Moreover, we survey orthogonality properties of these polynomials. Finally, by applying another method which is related to p-adic integrals, we derive some formulas and combinatorial sums associated with some well-known numbers and polynomials.

Purpose The purpose of this paper is to develop a loan insurance pricing model allowing for the skewness and kurtosis existing in underlying asset returns. Design/methodology/approach Using the theory of Gram-Charlier option, the authors first derive a closed-form solution of the Gram-Charlier pricing model. To address the difficulties in implementing the pricing model, the authors subsequently propose an iterative method to estimate skewness and kurtosis in practical application, which shows a relatively fast convergence rate in the empirical test. Findings Not only the theoretical analysis but also the empirical evidence shows that the effects of skewness and kurtosis on loan insurance premium tend to be negative and positive, respectively. Furthermore, the actual values of skewness and kurtosis are usually negative and positive, respectively, which leads to the empirical result that the pricing model ignoring skewness and kurtosis substantially underestimates loan insurance premium. Originality/value This paper proposes a loan insurance pricing model considering the skewness and kurtosis of asset returns, in which the authors use the theory of Gram-Charlier option. More importantly, the authors further propose a novel iterative method to estimate skewness and kurtosis in practical application. The empirical evidence suggests that the Gram-Charlier pricing model captures the information content of skewness and kurtosis.

In this paper, we propose a new method for image classification by the content in heterogeneous databases. This approach is based on the use of new series of separable discrete orthogonal moments as shape descriptors and the Support Vector Machine as classifier. In fact, the proposed descriptors moments are defined from the bivariate discrete orthogonal polynomials of Charlier-Meixner which are invariant to translation, scaling and rotation of the image. We also propose a new algorithm to accelerate the image classification process. This algorithm is based on two steps: the first step is the fast computation of the values of Charlier-Meixner polynomials by using a new recurrence relationship between the values of polynomials Charlier-Meixner. The second one is the new image representation and slice blocks. The proposed method is tested on three different sets of standard data which are well known to computer vision: COIL-100, 256-CALTECH and Corel. The simulation results show the invariance of the discrete orthogonal separable moments of Charlier-Meixner against the various geometric transformations and the ability for the classification of heterogeneous images.

In this paper, we suggest a new method of fast and stable calculation of the discrete orthogonal moments of Charlier and their inverses. This method is meant to accelerate the computation time and improve the quality of images reconstruction. In this method, we have combined two main concepts. The first concept is the digital filters based on the Z-transform to accelerate the calculation process of the discrete orthogonal moments of Charlier. The second concept is the partitioning of the image into a set of blocks of fixed sizes where each block is processed independently. The significant reduction in the image space during partitioning makes it possible to represent the minute details of the image with only low orders of Charlier’s discrete orthogonal moments, which ensures the digital stability during the processing of the image. In order to demonstrate the efficiency, stability, and precision of our method compared to other existing methods, some simulations have been performed on different types of binary images and gray images with and without noise.

The paper describes a method for resolving overlapping asymmetric peaks that make up a chromatogram. The presented method uses the Gram-Charlier model in the form of the first three terms of the Gram-Charlier series as a basis. Using the wavelet transform, the parameters of this model are determined, which is used to describe a single or overlapping chromatographic peak. Hermitian wavelets of the first four orders are used in the computation of the wavelet transform. To speed up the computation of multiple wavelet transforms, the possibility of coding a signal using the Chebyshev-Hermite functions is considered in order to further restore the set of wavelet transforms simultaneously. According to the presented method, the parameters of the peaks are determined by analytical expressions without using the numerical approximation of the chromatogram by the peak model, which avoids the disadvantages of the numerical approach. The resulting method is used to resolve overlapping asymmetric peaks. The advantage of the method over others is shown by calculating the area of each of the resolved peaks.

À l’heure où les politiques de l’enseignement supérieur et de la recherche célèbrent la pluri-, l’inter- ou la transdisciplinarité, plusieurs ouvrages de sociologie ou de science politique parus ces dernières années questionnent le sort des disciplines académiques. L’ouvrage collectif co-dirigé par Martin Benninghoff, Cécile Crespy, Jean-Émile Charlier et Jean-Philippe Leresche contribue à ce débat en analysant le « gouvernement des disciplines » que les éditeurs de ce volume définissent comm...

We present exact analytical expressions of moments of all orders for neuronal membrane potentials in the multiplicative nonstationary Poisson shot noise model. As an application, we derive closed-form Gram-Charlier density expansions that show how the probability density functions of potentials in such models differ from their Gaussian diffusion approximations. This approach extends the results of Brigham and Destexhe (Preprint, 2015a; Phys Rev E 91:062102, 2015b) by the use of exact combinatorial expressions for the moments of multiplicative nonstationary filtered shot noise processes. Our results are confirmed by stochastic simulations and apply to single- and multiple-noise-source models. Ministry of Education (MOE) This research is supported by the Ministry of Education, Singapore, under its Tier 1 Grant MOE2018-T1-001-201.

If a probability distribution is sufficiently close to a normal distribution, its density can be approximated by the truncated Gram-Charlier series where skewness and kurtosis directly appear as parameters. However, the existing literature is restricted to truncating the series expansion until the fourth moment because it becomes difficult to keep its non-negativity. This paper shows how the valid region of higher cumulants can be numerically implemented by the semi-definite algorithm, which ensures that a series truncated at a moment of arbitrary even order represents a valid probability density. Furthermore, the impact of higher moments on the valid regions has been shown.

Zeitschrift für Kristallographie / Crystalline materials 233(9-10), 695 - 706 (2018). doi:10.1515/zkri-2018-2060, The structure of glutathione, γ-l-Glutamyl-l-cysteinyl-glycine (C$_{10}$H$_{17}$N$_3$O$_6$S), was studied by multi-temperature single-crystal X-ray diffraction. Residual density maps from conventional independent atom model refinement gave indication of anharmonic motion in the molecule. This was further investigated by invariom refinement with anisotropic displacement parameters for all atoms, which described asphericity due to chemical bonding and lone pairs; afterwards only the residual-density signal of anharmonic motion remained. Treating anharmonicity with third-order Gram-Charlier displacement parameters led to regions with unphysical negative electron density. In contrast, a maximum entropy method (MEM) determination of the electron density successfully takes the features into account. Respective difference electron density plots (MEM minus prior and [Invariom+GC] minus invariom) agree well with each other. Challenges in treating and understanding the phenomenon are discussed. A procedure is proposed how unphysical negative electron density can be avoided. It is closely related to the free lunch algorithm., Published by De Gruyter, Berlin

International audience; Le mouvement de « retour aux sources » qui a caractérisé la théologie catholique durant la première moitié du XXe siècle est une réalité complexe, qui ne se laisse pas réduire à un schéma binaire opposant rénovateurs progressistes et leurs contradicteurs conservateurs. Le problème de la nature de la théologie spéculative telle que l’aurait comprise Thomas d’Aquin fournit un exemple certes inattendu et un peu marginal, mais néanmoins topique de la manière dont s’est opéré, en terre thomiste, le « retour aux sources ». Le dominicain belge Louis Charlier entreprit, durant les années 1930, de dégager la conception thomiste de la théologie spéculative des scories que des siècles de tradition des commentateurs y aurait indument accumulé au détriment de l’enseignement génuine de S. Thomas. Il ne cachait pas son inscription dans une ligne résolument rénovatrice. L’un de ses principaux critiques, le dominicain français Rosaire Gagnebet, nettement positionné dans les rivages romains, réputés conservateurs, du thomisme, réfuta sa thèse non par un alignement sur la position traditionnelle reçue des Commentateurs, mais par un retour encore plus rigoureux et radical à la véritable conception de la science qui était celle de S. Thomas. La thèse de Gagnebet fut d’ailleurs rapidement validée et adoptée par les plus réformateurs des thomistes adeptes de retour aux sources, tels Chenu ou Congar.

In this paper, we propose a new method for classification and Recognition of 3D Image Charlier moments using a Multilayer Perceptron architecture. The Multilayer Perceptron is an artificial neural network to evaluate the efficient structure in the non-linear systems. However, the determination of its architecture and weights is a fundamental issue due to their direct impact on the network convergence and performance. The robustness of the proposed approach have provided under many transforms. The experimental results show that our approaches are more robust than 3D Tchebichef moments.

In the present paper, we propose a certain integral modification of the operator, which involve Charlier polynomials with the weight function of generalized Baskakov and Szasz basis functions. We estimate some approximation properties and asymptotic formula for these operators. Also, the weighted approximation for these is given.

The purpose of this work is to give a form of positive linear operators involving Charlier polynomials, one of the discrete orthogonal polynomials which are generalization of Szasz operators. Also, Kantorovich type generalization of these operators are given. We obtain convergence properties of our operators with the help of Korovkin’s theorem and the order of approximation by using classical modulus of continuity.

International audience

In this contribution we consider the sequence $\{Q_{n}^{\lambda}\}_{n\geq 0} $ of monic polynomials orthogonal with respect to the following inner product involving differences \begin{equation*} \langle p,q\rangle _{\lambda}=\int_{0}^{\infty}p\left(x\right) q\left(x\right) d\psi ^{(a)}(x)+\lambda \,\Delta p(c)\Delta q(c), \end{equation*} where $\lambda \in \mathbb{R}_{+}$, $\Delta $ denotes the forward difference operator defined by $\Delta f\left(x\right) =f\left(x+1\right) -f\left(x\right) $, $\psi ^{(a)}$ with $a>0$ is the well known Poisson distribution of probability theory% \begin{equation*} d\psi ^{(a)}(x)=\frac{e^{-a}a^{x}}{x!}\quad \text{at}x=0,1,2,\ldots, \end{equation*}% and $c\in \mathbb{R}$ is such that $\psi ^{(a)}$ has no points of increase in the interval $(c,c+1)$. We derive its corresponding hypergeometric representation. The ladder operators and two different versions of the linear difference equation of second order corresponding to these polynomials are given. Recurrence formulas of five and three terms, the latter with rational coefficients, are presented. Moreover, for real values of $c$ such that $c+1, Comment: 1 Figure, Numerical Algorithms 2018

It has been shown in Ferreira et al. (Adv. Appl. Math 31:61–85, [2003]), Lopez and Temme (Methods Appl. Anal. 6:131–196, [1999]; J. Cpmput. Appl. Math. 133:623–633, [2001]) that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic expansions. In this paper we continue with that investigation and establish asymptotic connections between the fourth level and the two lower levels: we derive twelve asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Hermite, Charlier and Laguerre polynomials. From these expansions, several limits between polynomials are derived. Some numerical experiments give an idea about the accuracy of the approximations and, in particular, about the accuracy in the approximation of the zeros of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of the zeros of the Hermite, Charlier and Laguerre polynomials.

The Charlier polynomials are used for constructing oscillator-like systems. We introduce coherent states of the system are defined and study the main properties (in particular, the (over)completeness property). We show that the coherent states on the associated uncertainty relation take the minimum value. In the case under consideration, the Mandel parameter vanishes, which corresponds to the Poisson statistics of quasiexcitation spectrum for the considered oscillator. Bibliography: 30 titles.

The main historical purpose of the standard Gram-Charlier and Edgeworth series is to " correct" a Gaussian distributions when new information, such as moments, is given which does not match those of the Gaussian. These methods relate a probability distribution to a Gaussian by way of an operator transformation that is a function of the differentiation operator. We use the methods of the phase-space formulation of quantum mechanics to generalize these methods. We generalize in two ways. First, we relate any two probability distributions by way differentiation operator. Second, we generalize to the case where the differentiation operator is replaced by an arbitrary Hermitian operator. The generalization results in a unified approach for the operator transformation of probability densities. Also, when the Edgeworth and Gram-Charlier series are truncated, the resulting approximation is generally not manifestly positive. We present methods where the truncated series remain manifestly positive.

针对离散正交矩的平移和尺度不变量只能通过图像归一化或者借助几何矩不变量的线性组合间接获取，会带来较大的表示误差，本文基于离散Charlier多项式，提出了一种新的离散正交矩——Charlier矩，并给出了Charlier矩平移与尺度不变量的直接计算方法。实验表明，由Charlier矩直接计算的平移和尺度不变量与现有方法相比，具有较高的表示精度与分类准确率，而且对图像噪声有较强的稳定性，可以应用于图像不变分析与目标识别等应用领域。 The existing methods for extracting the translation and scale invariants from the discrete orthogonal moments are via a linear combination of the corresponding invariants of geometric moments or image normalization, which led to calculational errors. In this paper, a novel kind of discrete orthogonal moments named as Charlier moment is proposed based on the discrete Charlier polynomials, and then an approach to directly derive the translation and scale invariants from Charlier moments is also presented. Experimental results show the high classification and representation accuracy of these invariants as a result of direct calculation instead of the image normalization or a linear combination of the corresponding invariants of geometric moments. It is also shown that these invariants are relatively robust in the presence of image noise and are potentially useful as a kind of invariant descriptors in some image analysis and pattern recognition.

We propose a new approach to Poisson approximation. The basic idea is very simple and based on properties of the Charlier polynomials and the Parseval identity. Such an approach quickly leads to new effective bounds for several Poisson approximation problems. We also give a selected survey on diverse Poisson approximation results.

Moment-based features have been employed for solving various problems in pattern recognition and image processing. In this paper, we choose Charlier and Meixner moments separately, to solve texture segmentation, classification, and de-noising of images corrupted with Gaussian noise. In order to solve the texture classification problem, we propose to use lower-order Charlier and Meixner moments of 12 Haralick’s texture features and a two-class support vector machine. But, for texture segmentation problem, we use these moment-based texture energy features suggested for geometric moments by Mihran Tuceryan. Further, these energy features are used in K-means, Fuzzy K-means, and Kohonen’s neural network for solving texture segmentation problem, whereas for image de-noising problem, we use non-local mean (NLM) filter with Charlier and Meixner moments-based similarity values instead of pixel-based similarity values used in NLM for filter weights calculation. The simulation results show that the proposed applications worked well for two-class texture segmentation, classification of texture images and de-noising of images corrupted with a small amount of noise.

The present paper discusses a study of a class of Charlier matrix polynomials and its generalized analogue. Certain generating matrix functions, recurrence matrix relations, matrix differential equation, summation formulas and many new results have been discussed for these matrix polynomials. Weisner's group theoretic method is used to obtain matrix generating relations for Charlier matrix polynomials and the details of this method were given in this paper. Finally, we will discuss only briefly the procedure followed.

We derive uniform and non-uniform asymptotics of the Charlier polynomials by using difference equation methods alone. The Charlier polynomials are special in that they do not fit into the framework of the turning point theory, despite the fact that they are crucial in the Askey scheme. In this paper, asymptotic approximations are obtained respectively in the outside region, an intermediate region, and near the turning points. In particular, we obtain uniform asymptotic approximation at a pair of coalescing turning points with the aid of a local transformation. We also give a uniform approximation at the origin by applying the method of dominant balance and several matching techniques., Comment: 31 pages, 1 figure (in two separated files)

In this paper, we present a novel discrete orthogonal transformation called the separable Charlier-Krawtchouk transformation of fractional order (FrCKTs). This bivariate transformation is defined from the monovariable Charlier and Krawtchouk polynomials of fractional orders. Based on FrCKTs and the Sine Cosine Algorithm (SCA), a new method for the optimal reconstruction of large-size biomedical signals (bio-signals) is proposed. In addition, we propose a new technique for the zero watermarking of bio-signals for the protection of intellectual property rights. The results of simulations and comparisons clearly show the effectiveness of the proposed applications in the field of bio-signals.

Given a sequence of polynomials (pn)n, an algebra of operators A acting in the linear space of polynomials and an operator Dp ∈ A with Dp(pn) = npn, we form a new sequence of polynomials (qn)n by considering a linear combination of m consecutive pn: qn = pn + Pm j=1 βn,jpn−j . Using the concept of D-operator, we determine the structure of the sequences βn,j , j = 1, . . . , m, in order that the polynomials (qn)n are common eigenfunctions of an operator in the algebra A. As an application, from the classical discrete families of Charlier, Meixner and Krawtchouk we construct orthogonal polynomials (qn)n which are also eigenfunctions of higher order difference operators. Ministerio de Economía y Competitividad Junta de Andalucía

We introduce a new family of special functions, namely q-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to q-analogues of Poisson distributions. We focus our attention on their structural properties. Raising and lowering operators as well as Rodrigues-type formulas are obtained. An explicit representation in terms of a q-analogue of the second of Appell's hypergeometric functions is given. A high-order linear q-difference equation with polynomial coefficients is deduced. Moreover, we show how to obtain the nearest neighbor recurrence relation from some difference operators involved in the Rodrigues-type formula. The research of J. Arves u was partially supported by the research grant MTM2012-36732-C03-01 (Ministerio de Econom a y Competitividad) of Spain.

In this paper the Poisson-Charlier polynomials are introduced. Some of their recurrence relations are presented. Various families of bilinear and bilateral generating functions for these polynomials are derived. Furthermore, some special cases of the results are presented in this study. 2010 Mathematics Subject Classification: 33C45.

In this paper we prove in a constructing way that exceptional Charlier, Meixner, Hermite and Laguerre polynomials satisfy higher order recurrence relations. Our conjecture is that the recurrence relations provided in this paper have minimal order. Ministerio de Economía y Competitividad Junta de Andalucía Fondo Europeo de Desarrollo Regional

Rezension zu: Monique Dondin-Payre/Marie-Thérèse Raepsaet-Charlier (Ed.), Cités, municipes, colonies. Les processus de municipalisation en Gaule et en Germanie sous le Haut Empire romain. Histoire Ancienne et Médiévale, Vol. 53. Publications de la Sorbonne, Paris 1999. ISBN 2-85944-364-9, ISSN 0290-4500. XIX, 483 Seiten mit 2 Abbildungen, 13 Karten und zahlreichen Tabellen., Germania : Anzeiger der Römisch-Germanischen Kommission des Deutschen Archäologischen Instituts, Bd. 80 Nr. 1 (2002)

By using two operators representable by Jacobi matrices, we introduce a family of q-orthogonal polynomials, which turn out to be dual with respect to alternative q-Charlier polynomials. A discrete orthogonality relation and the completeness property for these polynomials are established.

Using Casorati determinants of Charlier polynomials View the MathML source, we construct for each finite set F of positive integers a sequence of polynomials View the MathML source, n∈σF, which are eigenfunctions of a second order difference operator, where σF is certain infinite set of nonnegative integers, σF⊊N. For suitable finite sets F (we call them admissible sets), we prove that the polynomials View the MathML source, n∈σF, are actually exceptional Charlier polynomials; that is, in addition, they are orthogonal and complete with respect to a positive measure. By passing to the limit, we transform the Casorati determinant of Charlier polynomials into a Wronskian determinant of Hermite polynomials. For admissible sets, these Wronskian determinants turn out to be exceptional Hermite polynomials.

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., Comment: 20 pages; minor changes

We derive asymptotics for the Lp-norms and information entropies of Charlier polynomials. The results differ to some extent from previously studied orthogonal polynomials, for example, the Lp-norms show a peculiar behaviour with two thresholds. Some complications arise because the measure involved is discrete.

The Pliocene hominin fossil 'Lucy' (A.L. 288-1