By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, the Christoffel-Darboux-type bilinear generating function and their asymptotic expressions, we obtain quadrature formulas for integral transforms generated by the classical orthogonal polynomials. These integral transforms, related to the so-called Poisson integrals, correspond to a modified Fourier Tra...

The finite Fourier transform of a family of orthogonal polynomials is the usual transform of these polynomials extended by 0 outside their natural domain of orthogonality. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families. 2010 Mathematics subject classification: primary 33C45; secondary 44A38, 33C47, 33C10, 42C10.

For a better understanding of the physical systems, even at quantum level, thermal quantities can be investigated. Recently, we realized that parity system also examined simultaneously, by substituting Dunkl operator with ordinary differential in mechanics. In this manuscript, consider two relativistic Dunkl-oscillators and investigate their well-known statistical methods. Besides, establish re...

with respect to the Gaussian measure. Macdonald extended it from An−1 to other root systems and verified his conjecture for classical ones by means of the Selberg integrals [M1]. It was established by Opdam in [O1] in full generality using the shift operators. The integral is an important normalization constant for a k-deformation of the Hankel transform introduced by Dunkl [D]. The generalized...

We study the Sobolev spaces of exponential type associated with the Dunkl-Bessel Laplace operator. Some properties including completeness and the imbedding theorem are proved. We next introduce a class of symbols of exponential type and the associated pseudodifferential-difference operators, which naturally act on the generalized Dunkl-Sobolev spaces of exponential type. Finally, using the theo...

To explore band structures of three-dimensional photonic crystals numerically, we need to solve the eigenvalue problems derived from the governing Maxwell equations. The solutions of these eigenvalue problems cannot be computed effectively unless a suitable combination of eigenvalue solver and preconditioner is chosen. Taking eigenvalue problems due to Yee’s scheme as examples, we propose using...

In this paper we prove the absolute continuity of the representing measures of the Dunkl intertwining operator and of its dual. Next we present some applications of this result. Key word : Dunkl intertwining operator and its dual. Absolute continuity of the representing measures. MSC (2000) : 33C80, 51F15, 44A15. Introduction We consider the differential-difference operators on R introduced by ...

Graphene-based materials are thought to revolutionize entire industries. Therefore, many research being carried on graphene theoretically and experimentally. On the other hand, recent studies show that use of Dunkl derivative, instead ordinary allows concept parity be interpreted together with physical quantities. In this manuscript, we investigate thermal quantities under constant magnetic fie...

Vector-valued Jack polynomials associated to the symmetric group SN are polynomials with multiplicities in an irreducible module of SN and which are simultaneous eigenfunctions of the Cherednik–Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r, p,N) and studi...

Orthogonality of the Jacobi and of Laguerre polynomials, P (α,β) n and L (α) n , is established for α, β ∈ C \ Z−, α + β 6= −2,−3, . . . using the Hadamard finite part of the integral which gives their orthogonality in the classical cases. Riemann-Hilbert problems that these polynomials satisfy are found. The results are formally similar to the ones in the classical case (when Rα,Rβ > −1).