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We introduce the analogue of Dunkl processes in the case of an affine root system of type ${\tilde{A}}_{1}$ . The construction of the affine Dunkl process is achieved by a skew-product decomposition by means of its radial part and a jump process on the affine Weyl group, where the radial part of the affine Dunkl process is given by a Gaussian process on the ultraspherical hypergroup $[0, 1]$ . We prove that the affine Dunkl process is a càdlàg Markov process as well as a local martingale, study its jumps, and give a martingale...

Affine Hecke algebras and orthogonal polynomials

I. G. MacDonald (1994/1995)

Séminaire Bourbaki

Affine Mappings And Elliptic Functions

G. D. Anderson, M. K. Vamanamurthy (1971)

Publications de l'Institut Mathématique

Affine mappings and elliptic functions.

G.D. Anderson, M.K. Vamanamurthy (1971)

Publications de l'Institut Mathématique [Elektronische Ressource]

Affine transformations of a Leonard pair.

Nomura, Kazumasa, Terwilliger, Paul (2007)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Algebraic properties of a family of Jacobi polynomials

John Cullinan, Farshid Hajir, Elizabeth Sell (2009)

Journal de Théorie des Nombres de Bordeaux

The one-parameter family of polynomials $J_{n} (x, y) = \sum_{j = 0}^{n} (\binom{y + j}{j}) x^{j}$ is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each $n \geq 6$ , the polynomial $J_{n} (x, y_{0})$ is irreducible over $ℚ$ for all but finitely many $y_{0} \in ℚ$ . If $n$ is odd, then with the exception of a finite set of $y_{0}$ , the Galois group of $J_{n} (x, y_{0})$ is $S_{n}$ ; if $n$ is even, then the exceptional set is thin.

Algebraic values of G-functions.

F. Beukers (1993)

Journal für die reine und angewandte Mathematik

Algèbres et noyaux de convolution sur le dual sphérique d’un groupe de Lie semi-simple, non compact et de rang $1$

M. Mizony (1976)

Publications du Département de mathématiques (Lyon)

Algorithm 47. Solution of a first-order non-linear differential equation in Chebyshev series

S. Lewanowicz (1976)

Applicationes Mathematicae

Allied integrals, functions, and series for the unit sphere.

Dzagnidze, O. (1998)

Georgian Mathematical Journal

Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier

Troels Roussau Johansen (2011)

Studia Mathematica

The maximal operator S⁎ for the spherical summation operator (or disc multiplier) $S_{R}$ associated with the Jacobi transform through the defining relation $\hat{S_{R} f} (λ) = 1_{| λ | \leq R} f ̂ (t)$ for a function f on ℝ is shown to be bounded from $L^{p} (ℝ ₊, d μ)$ into $L^{p} (ℝ, d μ) + L ² (ℝ, d μ)$ for (4α + 4)/(2α + 3) < p ≤ 2. Moreover S⁎ is bounded from $L^{p ₀, 1} (ℝ ₊, d μ)$ into $L^{p ₀, \infty} (ℝ, d μ) + L ² (ℝ, d μ)$ . In particular ${S_{R} f (t)}_{R > 0}$ converges almost everywhere towards f, for $f \in L^{p} (ℝ ₊, d μ)$ , whenever (4α + 4)/(2α + 3) < p ≤ 2.

Alternative orthogonal polynomials and quadratures.

Chelyshkov, Vladimir S. (2006)

ETNA. Electronic Transactions on Numerical Analysis [electronic only]

An Addition Theorem for Some q-Hahn Polynomials.

Charles F. Dunkl (1978)

Monatshefte für Mathematik

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