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This paper investigates time domain model order reduction of discrete-time bilinear systems with inhomogeneous initial conditions. The state of the system is approximated by the power series associated with the Charlier polynomials and the recurrence relation of the expansion coefficients is derived. The expansion coefficients are orthogonalized to construct the projection matrix by the modified multi-order Arnoldi method. The output of the resulting reduced order system maintains a certain number of expansion coefficients of the original output, and the error estimation of the reduced order system is briefly discussed. Due to the fact that the projection matrix involves the information of initial conditions, the proposed method can well reduce discrete-time bilinear systems with inhomogeneous initial conditions. Two numerical examples are employed to illustrate the effectiveness of the proposed method.

In this paper, we propose efficient methods for the reconstruction, compression, compressive sensing (CS) and encryption of 1D signals. The proposed reconstruction method is based on the use of Charlier moments (CMs) and the Artificial Bee Colony (ABC) algorithm. The latter is used for optimizing the local parameter of Charlier polynomials during the computation of CMs. In addition, new methods are presented for 1D signal compression and CS using CMs and ABC algorithm that guarantees a high quality of the decompressed/reconstructed signal. Moreover, we suggest a new signal encryption/decryption scheme relying on fractional-order Charlier moments and ABC algorithm, which is used for providing a high quality of the decrypted signal and for improving the security of the proposed scheme. The results of the conducted simulations and comparisons clearly show the efficiency of the proposed 1D-signal analysis methods.

The problem of 3D digital object invariability is encountered in image processing, especially in pattern classification/recognition. The 3D object should be correctly recognized regardless of its particular position and orientation in the scene. This paper proposes a new method to extract 3D Charlier moment invariants to translation and scaling (3DCMITS). These descriptors are extracted directly from discrete orthogonal Charlier polynomials without using 3D geometric moment invariants. This method is fast and does not require any numerical approximation compared to the indirect method based on 3D geometric moment invariants. The results show the proposed method's effectiveness in terms of speed with an improvement exceeding 99,97%. For validation purposes and as an illustration of the interest of 3DCMITS, this paper offers a classification system for 3D objects based on the proposed 3DCMITS and Support Vector Machine (SVM) classifier. The obtained results are verified with K-Nearest Neighbor (KNN) classifier and other existing works in the literature.

The main aim of this paper is to define and investigate more general multiple Charlier polynomials on the linear lattice $\omega \mathbb{N} = \{ 0,\omega ,2\omega ,\ldots \} $ ω N = { 0 , ω , 2 ω , … } , $\omega \in \mathbb{R}$ ω ∈ R . We call these polynomials ω-multiple Charlier polynomials. Some of their properties, such as the raising operator, the Rodrigues formula, an explicit representation and a generating function are obtained. Also an $( r+1 )$ ( r + 1 ) th order difference equation is given. As an example we consider the case $\omega =\frac{3}{2}$ ω = 3 2 and define $\frac{3}{2}$ 3 2 -multiple Charlier polynomials. It is also mentioned that, in the case $\omega =1$ ω = 1 , the obtained results coincide with the existing results of multiple Charlier polynomials.

In this paper, we propose a zero-watermarking scheme based on a new set of quaternion moments called Quaternion Radial Fractional Charlier Moments (QRFrCMs) for the copyright protection of color medical images. The proposed moments are developed from a new type of discrete orthogonal polynomials called Fractional Charlier Polynomials (FrCPs) and from quaternion theory. The robustness of the proposed scheme is ensured thanks to robustness of the proposed quaternion moments against image processing attacks and geometric attacks where the BCR is greater than 98%. In addition, the scheme uses the fractional order of QRFrCMs and a chaotic system based on two-dimensional CML (2DCML) using mixed linear-nonlinear coupling to provide a high level of security. Experimental results show the superiority of the proposed scheme over other recent schemes in terms of robustness against geometric attacks and common image processing attacks. The proposed scheme can be used for the secure transmission of color medical images over insecure networks.

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.

The computation of high-order discrete orthogonal moments is very complex and unstable due to fluctuations in numerical polynomial values. In this paper, we will present new algorithms for the stable computation of high-order discrete orthogonal Charlier polynomials and moments. These algorithms are based on the use of modified recurrence relations of Charlier polynomials with respect to the order n and with respect to the variable x, and on the use of the modified Gram-Schmidt orthonormalization process (GSOP). The algorithms in question allow the cancellation of terms that cause the numerical fluctuations of Charlier polynomial values during the recursive calculations. They also preserve the orthogonality property of high-order Charlier polynomials, which enables an efficient analysis of large-size bio-signals and 2D/3D images. Therefore, the simulation results prove the efficiency of the proposed algorithms when it comes to reconstructing large-size bio-signals and images.

In the present paper we introduce the B?zier variant of the Sz?sz-Durrmeyer type operators, involving the Poisson-Charlier polynomials. Our study focuses on a direct approximation theorem in terms of the Ditzian-Totik modulus of smoothness and the rate of convergence for differential functions whose derivatives are of bounded variation.

This paper discusses the applicability of data reconciliation approaches in metrology and mentions the existed shortcomings. The semi-parametric method based on Gram-Charlier series expansion is presented for overcoming the obstacles preventing the wider spread of the measured data reconciliation in metrological practice. The proposed approach allows the fast estimation of potential accuracy increase that matters for adaptive measurement systems. The corresponded expressions and tests are presented.

Bandwidth parameter estimation in univariate Kernel Density Estimation has traditionally two approaches. Rule(s)-of-Thumb (ROT) achieve ‘quick and dirty’ estimations with some specific assumption for an unknown density. More accurate solve-the-equation-plug-in (STEPI) rules have almost no direct assumption for the unknown density but demand high computation. This article derives a balancing third approach. Extending an assumption of Gaussianity for the unknown density to be estimated in \textit{normal reference} ROT (NRROT) to near Gaussianity, and then expressing the density using Gram-Charlier A (GCA) series to minimize the asymptotic mean integrated square error, it derives GCA series based Extended ROT (GCAExROT). The performance analysis using the simulated and the real datasets suggests to replace NRROT by a modified GCAExROT rule achieving a balancing performance by accuracy nearer to STEPI rules at computation nearer to NRROT, specifically at small samples.

Appell polynomials are certain family having wide range of application areas from numerical analysis to analytic function theory. They were defined by Paul Emile Appell in 1880. The most famous Appell polynomials are Hermite, Bernoulli and Euler polynomials. This thesis consists of five chapters. Chapter 1 is devoted to ntroduction. In Chapter 2, we give basic definitions and properties that is used throughout the thesis. Chapters 3, 4 and 5 are original. In Chapter 3, we introduce 3D-ω-Hermite Appell polynomials An (x, y,z;ω) using ω-Hermite polynomials Gω n (x, y,z). We obtain their explicit forms, determinantal representations, recurrence relations, lowering and raising operators, difference equations, integro-difference equations, and partial difference equations. In Chapter 4, we introduce the ∆ω-multiple Appell polynomials and we give an explicit representation and recurrence relations for them. In the last chapter, we define ω-multiple Charlier polynomials then we give raising operator, Rodrigues formula, explicit representation and generating function. Also an (r +1)th order difference equation is given. As an example we consider the case ω = 3 2 and define 3 2−multiple Charlier polynomials. Keywords: ∆ω-Appell polynomials, determinant, recurrence equation, Multiple orthogonal polynomials, ω- multiple Charlier polynomials, Appell polynomials, Hypergeometric function, Rodrigues formula, Generating function, Difference equation Doctor of Philosophy in Mathematics. Institute of Graduate Studies and Research. Thesis (Ph.D.) - Eastern Mediterranean University, Faculty of Arts and Sciences, Dept. of Mathematics, 2021. Supervisor: Prof. Dr. Mehmet Ali Özarslan.

We give new explicit representations as well as new generating functions for the associated Meixner, Charlier, Laguerre, and Krawtchouk polynomials. The obtained results are then used to derive new generating functions and convolution-type formulas of the corresponding classical polynomials. Some consequences of our results are also mentioned., 14 pages. arXiv admin note: substantial text overlap with arXiv:2103.08367

This paper is a sequel to Aschakulporn and Zhang (J Futures Mark 42(3):365–388, 2022). The errors of the Bakshi et al. (Rev Financ Stud 16(1):101–143, 2003) risk-neutral moment estimators is studied using the Gram–Charlier density—with the skewness and excess kurtosis specified. To obtain skewness with (absolute) errors less than $$10^{-3}$$ 10 - 3 , the range of strikes ($$K_{\min }, K_{\max }$$ K min , K max ) must contain at least 3/4 to 4/3 of the forward price and have a step size ($$\Delta K$$ Δ K ) of no more than 0.1% of the forward price. The range of strikes and step size corresponds to truncation and discretization errors, respectively. This is consistent to Aschakulporn and Zhang (2022) for non-volatile market periods.

This paper proposes risk measures for bank solvency by accurately measuring the solvency risk components. These measures consider the minimum regulatory solvency levels and banks’ risk appetite level and risk profile. For this purpose, we used semi-nonparametric statistics to model stylized facts of the risk distribution, particularly the high-order moments of the Solvency Decline Rate, the Tier Decline Rate, and the Portfolio Growth Rate variables. Additionally, these risk measures can be used to measure the risk of regulatory intervention and to define policies that establish the minimum solvency levels required by banking regulators by estimating the Quantile Risk Metrics. As a case study, we collected data on the solvency indicators of the Colombian banking system, which adapts to the standards established by the Basel Committee. According to the results, the liquidity injection measures implemented in response to the needs generated by the COVID-19 pandemic led to an increase in the levels of the risk portfolio in the Colombian banking system, which exceeded the 99th percentile of the probability distribution of monthly portfolio value changes.

Given a finite set of positive integers G and polynomials Rg, g∈G, with degree of Rg equal to g, we associate to them a sequence (qn(x))n of Charlier type polynomials defined from the Charlier poly...

This study is devoted to obtaining the Charlier series solutions of first order delay differential equations involving proportional and constant arguments by employing an inventive numerical method dependent upon a collaboration of matrix structures derived from the parametric Charlier polynomial. The method essentially conducts the conversion of the unknown terms into a unique matrix equation at the collocation points, which yields a direct computation for these stiff equations. Two illustrative examples are included to test the accuracy and efficiency of the method. According to the investigation of the graphical and numerical results, the method holds fast, inventive and accurate computation, regularizing the matrix forms in compliance with the equations in question.

The Gram–Charlier expansion of a target probability density, (Formula presented.), is an (Formula presented.) -convergent series (Formula presented.) in terms of a reference density (Formula presented.) and its orthonormal polynomials (Formula presented.). We implement this for the density of a regime-switching Lévy process at a given time horizon T. The main step is the evaluation of moments of all orders of (Formula presented.) in terms of model primitives, for which we give a matrix-exponential representation. A number of numerical examples, in part involving pricing of European options, are presented. The traditional choice of (Formula presented.) as normal with the same mean and variance as (Formula presented.) only works for the regime-switching Black–Scholes model. Outside the scope of Black–Scholes, (Formula presented.) is typically taken as a normal inverse Gaussian. A similar analysis is given for time-changed Lévy processes modelling stochastic volatility.

Model Black-Scholes merupakan salah satu model untuk menentukan harga opsi tipe Eropa dengan asumsi return dari saham harus berdistribusi normal. Pada kenyataannya return saham tidak mengikuti bentuk distribusi normal, dimana artinya skewness dan kurtosis saham tidak sama dengan 0 dan 3. Untuk mengatasi ketidaknormalan dari return saham dilakukan modifikasi model Black-Scholes dengan ekspansi Gram-Charlier dengan pendekatan polynomial untuk mengatasi bentuk data yang tidak berdistribusi normal. Harga opsi dengan ekspansi Gram-Charlier dipengaruhi oleh beberapa faktor yakni harga saham awal (), harga kontrak (), tingkat suku bunga bebas risiko (), waktu jatuh tempo (), volatilitas ( ), skewness dan kurtosis. Hasil dari studi menunjukkan bahwa dengan penentuan harga opsi dengan menggunakan metode Gram-Charlier sedikit lebih mendekati harga pasar dari pada menggunakan model Black-Scholes.Kata Kunci: Call option, Black-Scholes, Ekspansi Gram-Charlier, polinomial Hermite, kurtosis

This paper implements a procedure for dynamically selecting the Gram–Charlier approximation that best fits the empirical distribution of cryptocurrency returns at any point in time. The endogenous selection of the Gram–Charlier expansion length exploits its property for approximating frequency distributions through a flexible number of parameters that allows capturing changes at the tails provoked by new extreme events. The procedure is based on the differences between the cumulative distribution function of Gram–Charlier distributions with a particular focus on the fitting of the distribution left tail for risk assessment purposes. The method is tested through backtesting techniques for a group of major cryptocurrencies. The results show that the selection of the Gram–Charlier expansion order on the basis of cumulative distribution function dynamics, provides, in most cases, a significant improvement for conditional coverage compared to the use of fixed-order Gram–Charlier expansions. The method seems to be a useful tool for risk management purposes, especially for highly volatile assets such as cryptocurrencies.

Cet article a pour ambition de situer l’œuvre de Jean Dubois (1920-2015) et Françoise Dubois-Charlier (1942-2016) dans la lignée de ce qu’on pourrait appeler l’école française de grammaire distributionnelle et transformationnelle, qui découle des travaux de Z. Harris et dont les deux autres membres importants sont Maurice Gross et Gaston Gross. L’accent est également mis sur l’influence de leurs travaux sur la vie intellectuelle française des années 1960-1990, de par la variété et la puissance de leurs talents. En effet, leur domaine de recherche ne se limite pas à la syntaxe, il s’étend aussi à l’analyse du discours, à la morphologie, à la neurolinguistique, à la pédagogie du français, et surtout à la lexicologie et la lexicographie, y compris la lexicographie informatisée. This paper positions the work of Jean Dubois (1920-2015) and Françoise Dubois-Charlier (1942-2016) in the tradition of the "French school of distributional and transformational grammar" along the lines of Z. Harris. Other prominent members of this trend were Maurice Gross and Gaston Gross. We emphasize the variety and power of their talent and show their influence on the French intellectual life from the sixties through the nineties. Their research was not limited to syntax, it spanned discourse analysis, morphology, neurolinguistics, French language teaching, but more importantly, it dealt with lexicography, whether computer-based or not.

I. Thèse et articles Dubois-Charlier, F. (1970). Etude neurolinguistique de l’alexie « pure », contribution historique et analyse neurolinguistique d’un groupe de 14 alexiques. Thèse de 3ème cycle : psychologie, Paris 8. Dubois-Charlier, F. (1970). Eléments de linguistique anglaise : syntaxe. Collection « Langue et Langage », Paris : Larousse Dubois-Charlier, F. (1971). Eléments de linguistique anglaise : la phrase complexe et les nominalisations. Paris : Larousse Dubois-Charlier, F. (1971). ...

Notre article décrit la structure des ressources lexicales Les Verbes Français (LVF) et le Dictionnaire Électronique des Mots (DEM) élaborées pendant plusieurs années par Jean Dubois et Françoise Dubois-Charlier. Nous suggérons ensuite des utilisations possibles de ces ressources pour le traitement automatique de la langue (TAL). Compte-tenu du fait que LVF a déjà fait l'objet de plusieurs travaux au cours des dernières décennies, nous insistons sur le DEM, une ressource linguistique particulièrement mal connue qui peut être considérée comme la synthèse des travaux lexicographiques de Dubois et Dubois-Charlier. Le DEM souffre d’être resté inachevé, mais son extension peu commune (près de 150 000 entrées) et surtout ses corrélations avec LVF en font une source de données lexicales de premier ordre pour la linguistique du français et pour le TAL. Nous présentons de nouvelles versions du LVF et du DEM au format JSON avec une nouvelle interface de consultation de ces dictionnaires. Nous espérons de la sorte favoriser la diffusion de ces ressources lexicales auprès de la communauté des chercheurs en lexicologie, en lexicographie et en TAL. This paper presents the structure of two French lexical resources, Les Verbes Français (LVF) and Dictionnaire Électronique des Mots (DEM), created over many years by Jean Dubois and Françoise Dubois-Charlier. Applications of these resources for Natural Language Processing (NLP) are then sketched. Given the fact that LVF has already been studied quite extensively over the last decades, DEM is described in details as it can be considered as the synthesis of the lexicographic works of Dubois and Dubois-Charlier. DEM was unfortunately left as a work in progress, but its extension (almost 150 000 entries) and especially its links with LVF constitute an important French lexical resource for NLP applications. A new JSON based format of LVF and DEM is also presented with an integrated query interface. This should promote the use of these resources by researchers in lexicology, lexicography and NLP.

Françoise Dubois-Charlier a joué un rôle non négligeable dans le lancement de la rue LANGAGES par Jean Dubois en 1966. Elle a œuvré à l'élargissement du domaine des sciences du langage de deux façons : en fournissant une documentation détaillée et critique sur de nouvelles théories de l'interface entre syntaxe et sémantique en provenance des Etats-Unis, la sémantique générative (n°27, 1972) et la grammaire des cas (n°35, 1975) et en défendant la dimension neurolinguistique aux côtés de son mari et de Henry Hécaen (n°25, 1972, Neurolinguistique et neuropsychologie) et en coordonnant un numéro entier (n°44, 1976) à une pathologie peu connue, l'alexie, sujet de sa thèse, soutenue en 1970. Françoise Dubois-Charlier played a significant role in the creation of the journal LANGAGES by Jean Dubois in 1966. She worked to expand the field of language sciences in two ways: by providing detailed and critical documentation on new theories of the interface between syntax and semantics from the United States, generative semantics (n°27, 1972) and case grammar (n°35, 1975) and by defending the neurolinguistic dimension alongside her husband and Henry Hécaen (n°25, 1972, Neurolinguistique et neuropsychologie) and by editing an issue (n°44, 1976) on a little-known pathology, alexia, the topic of her PhD thesis, defended in 1970.

In this paper, we propose a new type of hybrid discrete orthogonal moments called the Tchebichef-Charlier (TC) moments. This type of moment is created on the basis of the Tchebichef and Charlier polynomials in order to exploit the important properties of each one of these polynomials. This paper also suggests a new fast method for the reconstruction of the large-size 1D signal. This proposed method requires the signal conversion from 1D to 2D space and then the use of matrix formulations for the reconstruction in the 2D space. The simulations performed clearly show the usefulness of the proposed hybrid TC moments for the reconstruction and for the localization of regions of interest (ROI) of a large-size 1D signal.

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers have been studied, and several integer sequences related to them have been introduced. In this article other types of Sheffer polynomials are considered, by introducing the adjoint Poisson-Charlier and the adjoint Related polynomials.

Krall–Charlier polynomials ( c n a ; F ) n are orthogonal polynomials which are also eigenfunctions of a higher order difference operator. They are defined from a parameter a (associated to the Charlier polynomials) and a finite set F of positive integers. We study the algebra D a F formed by all difference operators with respect to which the family of Krall–Charlier polynomials ( c n a ; F ) n are eigenfunctions. Each operator D ∈ D a F is characterized by the so called eigenvalue polynomial λ D : λ D is the polynomial satisfying D ( c n a ; F ) = λ D ( n ) c n a ; F . We characterize the algebra of difference operators D a F by means of the algebra of polynomials D a F = { λ ∈ C [ x ] : λ ( x ) = λ D ( x ) , D ∈ D a F } . We associate to the family ( c n a ; F ) n a polynomial Ω F a and prove that, except for degenerate cases, the algebra D a F is formed by all polynomials λ ( x ) such that Ω F a divides λ ( x ) − λ ( x − 1 ) . We prove that this is always the case for a segment F (i.e., the elements of F are consecutive positive integers), and conjecture that it is also the case when the Krall–Charlier polynomials ( c n a ; F ) n are orthogonal with respect to a positive measure.

Cet article constitue un bref aperçu du parcours de Françoise Dubois-Charlier, resituant ses travaux dans leur contexte historique, depuis ses débuts en psycholinguistique jusqu’aux dictionnaires électroniques (Les verbes français), en passant bien sûr et avant tout par la grammaire générative et transformationnelle, la sémantique générative, les structuralisme, mais aussi la linguistique de corpus. This paper provides an overview of Françoise Dubois-Charlier’s career, placing her works in their historical context, from her early career in psycholinguistics to her latest works on electronic dictionaries (Les verbes français in this paper), including of course and above all generative and transformational grammar, generative semantics, structuralism, but also corpus linguistics.

Antoinette Balibar-Mrabti Comment évoquer le couple d’enseignants-chercheurs que formèrent Jean Dubois et Françoise Dubois-Charlier ? L’excellente photographie en couleur choisie en 2017 pour le colloque d’Aix en Provence qui leur a été consacré suffirait déjà parfaitement pour nous rappeler l’impression qu’ils produisaient, celle d’un type d’universitaires, d’un abord exigeant et même strict, que soulignait une élégance vestimentaire discrète. Une alchimie d’intelligence et de goûts aura per...

In this paper we study an extension of the Gram–Charlier (GC) density in Jondeau and Rockinger (2001) which consists of a Gallant and Nychka (1987) transformation to ensure positivity without parameter restrictions. We derive its parametric properties such as unimodality, cumulative distribution, higher-order moments, truncated moments, and the closed-form expressions for the expected shortfall (ES) and lower partial moments. We obtain the analytic th order stationarity conditions for the unconditional moments of the TGARCH model under the transformed GC (TGC) density. In an empirical application to asset return series, we estimate the tail index; backtest the density, VaR and ES; and implement a comparative analysis based on Hansen’s skewed-t distribution. Finally, we present extensions to time-varying conditional skewness and kurtosis, and a new class of mixture densities based on this TGC distribution. Financial support from the Spanish Ministry of Economy and Competitiveness, Spain through grant ECO2017-87069-P is gratefully acknowledged by Ángel León.

In this paper, we give some direct approximation results by modified Kantorovich–Szász type operators involving Charlier polynomials. Further, approximation results are also developed in polynomial weighted spaces. Moreover, for the functions of bounded variation, approximation results are proved. Finally, some graphical examples are provided to show comparisons of convergence between old and modified operators towards a function under different parameters and conditions.

With the analysis of a textbook case, the inflectional category of dual in contemporary French, this article presents the hypothesis of a rise in morphology among the founding disciplines of grammar in written languages. Through a study of morphographies, this trend is considered here as a result of the emergence and development of so-called electronic dictionaries, with their lexicographical words as entries to access form / meaning associations. We know that these dictionaries, piloted by mixed teams of computer scientists and linguists, impose themselves step by step as major classificatory tools for the most general treatments, in theoretical and applied linguistics, now related in our modernity to the exploitation of large corpora that have become digitised.

The paper deals with the approximation of first order derivative of a function by the first order derivative of Szász-type operators based on Charlier polynomials introduced by Varma and Taşdelen [20]. The uniform convergence theorem, Voronovskaja type asymptotic theorem and an estimate of error in terms of the second order modulus of continuity of the derivative of the function are investigated. Further, it is shown that linear combinations of the derivative of the above operators converge to the derivative of function at a faster rate. Finally, an estimate of error in the approximation is obtained in terms of the \begin{document}$ (2k+2)th $\end{document} order modulus of continuity using Steklov mean.

We introduce a new sequence of linear positive operators by combining the Charlier polynomials and the Szasz type operators defined by Jain (in J Aust Math Soc 13(3):271–276, 1972). We obtain the degree of approximation of these operators by means of the weighted modulus of continuity and for functions in a Lipschitz type space. We also study the approximation of functions having a derivative equivalent with a function of bounded variation.

The aim of the paper is to study the characterization of delta operator associated with some Sheffer polynomials. In this paper, we consider Poisson-Charlier polynomials and investigate the characterization of delta operator via sequential representation of delta operator. From our investigation, we are able to prove an interesting propositions for the above mentioned.

We analyze the asymptotic distribution of roots of Charlier polynomials with negative parameter depending linearly on the index. The roots cluster on curves in the complex plane. We determine implicit equations for these curves and deduce the limiting density of the root distribution supported on these curves. The proof is based on a determination of the limiting Cauchy transform in a specific region and a careful application of the saddle point method. The obtained result represents a solvable example of a more general open problem., 26 pages

Nathan Charlier est docteur en sciences politiques et sociales et chercheur au Département des Sciences de la Santé publique au sein de l’Université de Liège (Belgique). Ses recherches en sociologie de l’action publique le conduisent fréquemment à analyser les interactions entre savoirs et pouvoirs. L’auteur examine dans cet ouvrage la manière dont, au sein de l’état fédéral belge, deux entités fédérées, la Wallonie et la Flandre, ont adopté au cours des dernières décennies des trajectoires d...

In the present paper, we introduce a generalized Szasz type operators based on ´ ρ(x) where ρ is a continuously differentiable function on [0, ∞), ρ(0) = 0 and inf ρ 0 (x) ≥ 1, x ∈ [0, ∞). This function not only characterizes the operators but also characterizes the Korovkin set 1, ρ, ρ2 in a weighted function space. First, we establish approximation in a Lipschitz type space and weighted approximation theorems for these operators. Then we obtain a Voronovskaja type result and the rate of convergence in terms of the weighted modulus of continuity.

Il s’agit ici de convier à la lecture de L’héritage de Jean Dubois et Françoise Dubois-Charlier toute personne cherchant un descriptif des orientations et productions qui furent celles de ces deux linguistes, avec leurs analyses et leurs résultats, et l’impressionnante influence qu’ils ont pu exercer sur les chercheurs. Et encore, comme il est précisé dans la « Présentation du numéro », il demeure des domaines de préoccupation des deux auteurs non abordés. ...

Simsek, Burcin/0000-0003-2857-6629; SIMSEK, YILMAZ/0000-0002-0611-7141; KUCUKOGLU, IREM/0000-0001-9100-2252 WOS: 000505589700029 The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson-Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution. Scientific Research Project Administration of Akdeniz UniversityAkdeniz University This paper is dedicated to Hari Mohan Srivastava on the occasion of his 80th Birthday. Yilmaz Simsek was supported by the Scientific Research Project Administration of Akdeniz University.