The aim of this paper is to construct sup-exponentially localized kernels and frames in the context of classical orthogonal expansions, namely, expansions in Jacobi polynomials, spherical harmonics, orthogonal polynomials on the ball and simplex, and Hermite and Laguerre functions.

We consider determinants of Wronskian type whose entries are multiple orthogonal polynomials associated with a path connecting two multi-indices. By assuming that the weight functions form an algebraic Chebyshev (AT) system, we show that the polynomials represented by the Wronskians keep a constant sign in some cases, while in some other cases oscillatory behavior appears, which generalizes cla...

Ever since Kaplansky and Riordan [9-l I] developed their elegant theory of permutations with restricted position, a great variety of combinatorial problems have been explicitly solved, for example, see Dillon and, Roselle [l], Goldman, Joichi, and White [4-S], and Foata, Schutzenberger [2; 31. Spurred by the oft-noted analogy between the case of finite boolean algebras and that of lattices of v...

Linear operators which (1) preserve the reality of zeros of polynomials having only real zeros and (2) map stable polynomials into stable polynomials are investigated using recently established results concerning the zeros of certain Fox–Wright functions and generalized Mittag-Leffler functions. The paper includes several open problems and questions.  2005 Elsevier Inc. All rights reserved.

1. Fourier transform on Rn 1 2. Fourier analysis on the Heisenberg group 2 2.1. Representations of the Heisenberg group 2 2.2. Group Fourier transform 3 2.3. Convolution and twisted convolution 5 3. Hermite and Laguerre functions 6 3.1. Hermite polynomials 6 3.2. Laguerre polynomials 9 3.3. Special Hermite functions 9 4. Group Fourier transform of radial functions on the Heisenberg group 12 Ref...

We present the exact solution for a set of nonlinear algebraic equations 1 zl = πd + 2d n ∑ m6=l 1 zl−zm . These were encountered by us in a recent study of the low energy spectrum of the Heisenberg ferromagnetic chain [1]. These equations are low d (density) “degenerations” of more complicated transcendental equation of Bethe’s Ansatz for a ferromagnet, but are interesting in themselves. They ...

An important chapter in the theory of distribution of zeros of polynomials and transcendental entire functions pertains to the study of linear operators acting on entire functions. This article surveys some recent developments (as well as some classical results) involving some specific classes of linear operators called multiplier sequences and complex zero decreasing sequences. This expository...

An efficient algorithm and a Fortran 90 module (LaguerrePol) for computing Laguerre polynomials L (α) n (z) are presented. The standard threeterm recurrence relation satisfied by the polynomials and different types of asymptotic expansions valid for n large and α small, are used depending on the parameter region. Based on tests of contiguous relations in the parameter α and the degree n satisfi...