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In this paper, we establish a distributional Rodrigues formula for non-symmetric Dunkl-classical orthogonal polynomial sequences. Then, we use this formula to determine, explicitly, the coefficients in the three-term recurrence relation that non-symmetric Dunkl-classical polynomial sequences satisfy.

(2017). WITHDRAWN: Classification of nonsymmetric Dunkl-classical orthogonal polynomials. Journal of Difference Equations and Applications: Vol. 23, No. 7, pp. 1143-1160.

In this paper, we classify nonsymmetric Dunkl-classical linear functionals. Firstly, we reduce the characterization of a Dunkl-classical linear functional given by the first author in a previous work to a Tμ-distributional equation of Pearson’s type. Secondly, after rescaling the parameters, we prove that the unique nonsymmetric Dunkl-classical linear functional is the perturbed generalized Gegenbauer form.

In this paper, we present three characterizations of Dunkl-classical orthogonal polynomials. The first one is a -distributional equation of Pearson type fulfilled by its associated form, where is the Dunkl operator. The second is a first order linear -difference equation with polynomial coefficients satisfied by the corresponding Stieltjes function. The third is the so-called structure relation fulfilled by Dunkl–classical polynomials.

In this paper, we deal with some linear functionals on the vector space of polynomials whose moments are, in certain normalization, classical orthogonal polynomials (Hermite, Laguerre and Gegenbauer). We show that these linear functionals are semiclassical of class, at most, three. We give the coefficients in the three-term recurrence relations that the corresponding monic orthogonal polynomial sequences satisfy.

We deal with monic orthogonal polynomial sequences (MOPSs), , satisfying the three-term recurrence relation , , with initial conditions and , where and for all . These sequences are characterized by the relation , where is a MOPS. In this paper, we show that the sequence is -semiclassical if and only if the sequence is -semiclassical. Then, we express the characteristic elements of the -semiclassical sequence , such as the q-Pearson equation satisfied by the corresponding linear functional, the class of the linear functional, the first-order linear q-difference equation satisfied by the Stieltjes function and the coefficients of the structure relation for such a sequence of polynomials, in terms of the characteristic elements of the sequence, . In particular, if the sequence is -semiclassical of class zero, then we obtain a new non-symmetric -semiclassical sequence of polynomials of class s = 1.

We evaluate the determinants of Hankel matrices, whose elements are a linear combination of three successive shifted Catalan numbers. This is done by finding a Jacobi linear functional, such that their moments are, up to a multiplicative constant, the Catalan numbers. The values of such determinants are then expressed in terms of Jacobi polynomials.

An orthogonal polynomial sequence with respect to a regular form (linear functional) u is said to be semi-classical if there exist a monic polynomial Φ and a polynomial Ψ, with deg Ψ≥1, such that (Φ u)′+Ψ u=0. Recently, all semi-classical monic orthogonal polynomial sequences of class one satisfying a three-term recurrence relation with β n =(−1) n β0, n≥0, β0∈ℂ∖{0} have been determined (see [B. Bouras and A. Alaya, A large family of semi-classical polynomials of class one, Integral Transforms Spec. Funct. 18 (2007), pp. 913–931]). In this paper, the sequences of the above family such that their corresponding Stieltjes function S(u)(z)=−∑ n≥0⟨ u, x n ⟩/z n+1 satisfies a quadratic relation of the form BS 2(u)+CS(u)+D=0, where B, C, D are polynomials, are described.

In [4] the authors proved that a quasi-symmetric orthogonal polynomial sequence $${{\{R_{n}\}}_{n \ge 0}}$$ is a Laguerre-Hahn sequence if and only if the component $${{\{P_{n}\}}_{n \ge 0}}$$ in its quadratic decomposition is also a Laguerre-Hahn sequence. In this paper and under these conditions, we deduce the class s of the Laguerre-Hahn sequence $${{\{R_{n}\}}_{n \ge 0}}$$ . More precisely, if s′ is the class of $${{\{P_{n}\}}_{n \ge 0}}$$ then 2s′ ≤ s ≤ 2s′ + 3. On the other hand the polynomial coefficients of the Riccati equation satisfied by the Stieltjes function corresponding to $${{\{R_{n}\}}_{n \ge 0}}$$ are given in terms of those of $${{\{P_{n}\}}_{n \ge 0}}$$ . As an application, we determine all non-symmetric quasi-symmetric Laguerre-Hahn sequences of class one.

The aim of this work is to study the family of semi-classical sequences of polynomials fulfilling the three-term recurrence relation with β n =(−1) n β0, n≥0, where . Some information about the shape of polynomials intervening in the distributional equation in which the corresponding linear functional (or form) satisfies are given, and two identities satisfied by the appropriate structure relation coefficients are established. These results are determinative to work out in detail the particular case when β0≠0 and the class is equal to one. In this case, we give explicitly the recurrence coefficients, the structure relation coefficients and an integral representation.

In this paper, by evaluating a new Hankel determinant of order n, we give explicitly the recurrence coefficients of a symmetric semi-classical sequence of polynomials of class s ≤ 2. Then, by using the quadratic decomposition of such sequence, we give a nonsymmetric semi-classical sequence of polynomials of class generalizing the Jacobi polynomial sequence.

We give, explicitly the recurrence coefficients, integral representations and the structure relation coefficients of two semi-classical sequences of polynomials: the first is symmetric of class s≤2 generalizing the generalized Hermite sequence, and the second is nonsymmetric of class equal or less than one.

In this paper, we show that Dunkl-classical polynomial sequences can be characterized by a differential-difference equation. AMS Subject Classification: 33C45, 42C05

The present contribution describes two prediction methods for flows around transonic airfoils, including shock control devices. The whole work was done in the frame of the European Shock Control Investigation Project EUROSHOCK-AER-2, and the global objective was the improvement of the flight performance, in transonic speed, in terms of cruise speed, fuel consumption and exhaust emissions for both laminar and turbulent wings. More specifically the “passive” control of shock/boundary layer interaction, whereby part of the solid surface of the airfoil is replaced by a porous surface over a shallow cavity, has been shown to be a means of improving the aerodynamic characteristics of supercritical airfoils.

A blade design and optimization procedure is presented in this work, which is based on viscous flow considerations. This procedure concerns the design of optimum rotating arbitrary compressible high subsonic compressor and turbine blade shapes. It takes into account the effects of wall curvature and Coriolis force on turbulence, while it allows the variation of stream surface radius, along which the blade shape is placed, as well as streamtube width, with meridional distance. In order to establish the inverse part of the viscous optimization procedure, aspects such as laminar stability, transition, optimum deceleration and, more generally, the behaviour of compressible attached and separated shear layers are discussed. A plane on which all the general properties of the compressible laminar and turbulent shear layers appear, is constructed and the generation of optimum shear layers for the critical side of the blade shape is established. The complete optimization (design) procedure is then described and discussed, while various designs realized by the present procedure are presented at the end of this paper.

Background information is presented on an arbitrary blade section design method which is outlined in a joint paper. This information concerns the assumptions, the development, and the predictive capabilities of the viscous flow calculation tool used in the design procedure. General properties of laminar and turbulent, unseparated or separated compressible shear layers, necessary for the blade optimization procedure, are discussed.

The advantages of acoustooptic techniques over classical electronic ones for signal correlation are reviewed. The limitations of these techniques and their drawbacks as compared to electronic systems are analyzed. By an appropriate association of the two systems it is possible to retain the advantages of both electronic and acoustooptic processing. The solution consists of making an appropriate interface between an acoustooptic correlator for rapid acquisition and tracking electronics which track the signal in a continuous way which is not possible in the acoustooptic correlator. This interface allows both acquisition and tracking in quasi-real time. >

We propose in this paper an acousto-optic processing unit for a rapicf Doppler resolving in the GPS system. A 1023-point acousto-optic correlator is reviewed and matched to our application through an adequate sampling and time-compressing of the GPS received sequences. Doppler resolving is implemented with a single frequency sweep loop. The systsem potentialities are compared to those of a classical one's. A processing time-gain up to 1000 is obtained.