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Canonical bases for 𝔰𝔩 ( 2 , ℂ ) -modules of spherical monogenics in dimension 3

Roman Lávička (2010)

Archivum Mathematicum

Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as 𝔰𝔩 ( 2 , ℂ ) -modules. As finite-dimensional irreducible 𝔰𝔩 ( 2 , ℂ ) -modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Gürlebeck in [3]. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.

Fractional Integration of the Product of Bessel Functions of the First Kind

Kilbas, Anatoly, Sebastian, Nicy (2010)

Fractional Calculus and Applied Analysis

Dedicated to 75th birthday of Prof. A.M. Mathai, Mathematical Subject Classification 2010:26A33, 33C10, 33C20, 33C50, 33C60, 26A09Two integral transforms involving the Gauss-hypergeometric function in the kernels are considered. They generalize the classical Riemann-Liouville and Erdélyi-Kober fractional integral operators. Formulas for compositions of such generalized fractional integrals with the product of Bessel functions of the first kind are proved. Special cases for the product of cosine...

On nodal sets and nodal domains on S 2 and ℝ 2

Alexandre Eremenko, Dmitry Jakobson, Nikolai Nadirashvili (2007)

Annales de l’institut Fourier

We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on S 2 . We also construct a solution of the equation Δ u = u in ℝ 2 that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.

Sidon sets and Riesz sets for some measure algebras on the disk

Olivier Gebuhrer, Alan Schwartz (1997)

Colloquium Mathematicae

Sidon sets for the disk polynomial measure algebra (the continuous disk polynomial hypergroup) are described completely in terms of classical Sidon sets for the circle; an analogue of the F. and M. Riesz theorem is also proved.

Some relations satisfied by Hermite-Hermite matrix polynomials

Ayman Shehata, Lalit Mohan Upadhyaya (2017)

Mathematica Bohemica

The classical Hermite-Hermite matrix polynomials for commutative matrices were first studied by Metwally et al. (2008). Our goal is to derive their basic properties including the orthogonality properties and Rodrigues formula. Furthermore, we define a new polynomial associated with the Hermite-Hermite matrix polynomials and establish the matrix differential equation associated with these polynomials. We give the addition theorems, multiplication theorems and summation formula for the Hermite-Hermite...

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