AUTOMORPHISMS OF ALGEBRAS AND ORTHOGONAL POLYNOMIALS

Suitable automorphisms together with complete classification of representations of some algebras can be used to generate some sets of orthogonal polynomials “at no cost”. This is also the case of nonstandard Klimyk-Gavrilik deformation Uq'(so3) which is connected to q-Racah polynomials

AUTOMORPHISMS OF ALGEBRAS AND ORTHOGONAL POLYNOMIALS

Suitable automorphisms together with complete classification of representations of some algebras can be used to generate some sets of orthogonal polynomials “at no cost”. This is also the case of nonstandard Klimyk-Gavrilik deformation Uq'(so3) which is connected to q-Racah polynomials

NOTE ON VERMA BASES FOR REPRESENTATIONS OF SIMPLE LIE ALGEBRAS

We discuss the construction of the Verma basis of the enveloping algebra and of finite dimensional representations of the Lie algebra An. We give an alternate proof of so-called Verma inequalities to the one given in [1] by P. Littelmann

Three variable exponential functions of the alternating group

New class of special functions of three real variables, based on the alternating subgroup of the permutation group $S_3$, is studied. These functions are used for Fourier-like expansion of digital data given on lattice of any density and general position. Such functions have only trivial analogs in one and two variables; a connection to the $E-$functions of $C_3$ is shown. Continuous interpolation of the three dimensional data is studied and exemplified.Comment: 10 pages, 3 figure

THREE-VARIABLE ALTERNATING TRIGONOMETRIC FUNCTIONS AND CORRESPONDING FOURIER TRANSFORMS

The common trigonometric functions admit generalizations to any higher dimension, the symmetric, antisymmetric and alternating ones. In this paper, we restrict ourselves to three dimensional generalization only, focusing on alternating case in detail. Many specific properties of this new class of special functions useful in applications are studied. Such are the orthogonalities, both the continuous one and the discrete one on the 3D lattice of any density, corresponding discrete and continuous Fourier transforms, and others. Rapidly increasing precision of the interpolation with increasing density of the 3D lattice is shown in an example