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Clemens Markett

Bio: Clemens Markett is an academic researcher from RWTH Aachen University. The author has contributed to research in topics: Differential equation & Orthogonal polynomials. The author has an hindex of 7, co-authored 25 publications receiving 191 citations.

Papers
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TL;DR: In this article, a new constructive approach is given to the linearization formulas of symmetric orthogonal polynomials, where an auxiliary function of four integer variables is introduced, which may be seen as a discrete analogue of Riemann's function.
Abstract: A new constructive approach is given to the linearization formulas of symmetric orthogonal polynomials. We use the monic three-term recurrence relation of an orthogonal polynomial system to set up a partial difference equation problem for the product of two polynomials and solve it in terms of the initial data. To this end, an auxiliary function of four integer variables is introduced, which may be seen as a discrete analogue of Riemann's function. As an application, we derive the linearization formulas for the associated Hermite polynomials and for their continuousq-analogues. The linearization coefficients are represented here in terms of3F2 and3Φ2 (basic) hypergeometric functions, respectively. We also give some partial results in the case of the associated continuousq-ultraspherical polynomials.

24 citations

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TL;DR: In this paper, it was shown that the Fourier-Bessel functions possess a product formula, the kernel of which splits up into two different parts, the first part is the well-known kernel of Sonine's product formula of Bessel functions, the second part is new and reflects the boundary constraints of the Fouriers-Borschtein differential equation.
Abstract: One of the most far-reaching qualities of an orthogonal system is the presence of an explicit product formula. It can be utilized to establish a convolution structure and hence is essential for the harmonic analysis of the corresponding orthogonal expansion. As yet a convolution structure for Fourier-Bessel series is unknown, maybe in view of the unpractical nature of the corresponding expanding functions called Fourier-Bessel functions. It is shown in this paper that for the half-integral values of the parameter\(\alpha = n + \frac{1}{2}\),n=0, 1, 2,⋯, the Fourier-Bessel functions possess a product formula, the kernel of which splits up into two different parts. While the first part is still the well-known kernel of Sonine's product formula of Bessel functions, the second part is new and reflects the boundary constraints of the Fourier-Bessel differential equation. It is given, essentially, as a finite sum over triple products of Bessel polynomials. The representation is explicit up to coefficients which are calculated here for the first two nontrivial cases\(\alpha = \frac{3}{2}\) and\(\alpha = \frac{5}{2}\). As a consequence, a positive convolution structure is established for\(\alpha \in \{ \frac{1}{2},\frac{3}{2},\frac{5}{2}\}\). The method of proof is based on solving a hyperbolic initial boundary value problem.

19 citations

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TL;DR: The Bessel-type functions as discussed by the authors are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity on the point of infinity of the complex plane, which are derived by linear combinations and limit processes from the classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials.
Abstract: The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane. There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation. When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coeffic...

16 citations

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TL;DR: In this article, the product formula for generalized and suitably normalized Hermite polynomials with parameter μ ≥ 0 was explicitly established and a generalized translation operator and a corresponding convolution product on appropriately weighted Lebesgue spaces were introduced.

14 citations


Cited by
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01 Jun 1893-Nature
TL;DR: The theory of functions is the basis on which the whole of pure mathematics which deals with continuously varying quantity rests as mentioned in this paper, and the answer would not be too wide nor would it always imply too much.
Abstract: WHAT is the theory of functions about? This question may be heard now and again from a mathematical student; and if, by way of a pattial reply, it be said that the elements of the theory of functions forms the basis on which the whole of that part of pure mathematics which deals with continuously varying quantity rests, the answer would not be too wide nor would it always imply too much. Theory of Functions of a Complex Variable. By Dr. A. R. Forsyth. (Cambridge University Press, 1893.)

491 citations

Journal ArticleDOI
Paul Neval1
TL;DR: In this paper, the authors show that the convergence and absolute convergence of orthogonal polynomials on infinite intervals and on the untt crrcle can be explained by the convergence of Christoffel functions.

372 citations

Journal ArticleDOI
TL;DR: It is shown that the global Galerkin matrix associated with complete polynomials cannot be diagonalized in the stochastically linear case.
Abstract: We investigate the structural, spectral, and sparsity properties of Stochastic Galerkin matrices as they arise in the discretization of linear differential equations with random coefficient functions. These matrices are characterized as the Galerkin representation of polynomial multiplication operators. In particular, it is shown that the global Galerkin matrix associated with complete polynomials cannot be diagonalized in the stochastically linear case.

133 citations

Journal ArticleDOI
TL;DR: This paper shall mainly deal with parameter augmentation forq-integrals such as the Askey?Wilson integral, the Nassrallah?Rahman Integral, theq-Integral form of Sears transformation, and Gasper's formula of the extension of the As key?Roy integral.

98 citations

Journal ArticleDOI
TL;DR: In this article, the authors constructed particular solutions for some heat transport differential equations, in particular, for extended forms of hyperbolic heat equation and of Guyer-Krumhansl (GK) equation.

57 citations