Bernoulli polynomials and Pascal matrices in the context of Clifford analysis

Malonek, Helmuth R.; Tomaz, Graça

doi:10.1016/j.dam.2008.06.009

Cited by 14 publications

(16 citation statements)

References 13 publications

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“…Next, one will look further for the construction of new classes of hypercomplex polynomials of discrete variable based on a Rodrigues-type representation of the operational formula (23). That consists into the construction of an operator σ(D + h ) that intertwines the Fischer duals Λ h = X h − log λ(D + h ), x and X h of D + h .…”

Section: Classes Of Discrete Hypercomplex Polynomialsmentioning

confidence: 99%

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

Faustino

2014

Applied Mathematics and Computation

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With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature (0, n) the umbral calculus framework with Lie-algebraic symmetries. The exponential generating function (EGF) carrying the continuum Dirac operator D = n j=1 e j ∂ xj together with the Lie-algebraic representation of raising and lowering operators acting on the lattice hZ n is used to derive the corresponding hypercomplex polynomials of discrete variable as Appell sets with membership on the space Clifford-vector-valued polynomials. Some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the Poisson-Charlier polynomials are introduced. Certain applications from the view of interpolation theory and integral transforms are also discussed.

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Section: Classes Of Discrete Hypercomplex Polynomialsmentioning

confidence: 99%

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

Faustino

2014

Applied Mathematics and Computation

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“…Several approaches for finding systems of solutions associated with discretized versions of (1) through combinatorics (cf. [8,9]), Liealgebraic representations (cf. [3,10,11]), and a combination of both (cf.…”

Section: Introductionmentioning

confidence: 99%

A conformal group approach to the Dirac–Kähler system on the lattice

Faustino

2017

Math Methods in App Sciences

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Abstract. Starting from the representation of the (n − 1) + n−dimensional Lorentz pseudo-sphere on the projective space PR n,n , we propose a method to derive a class of solutions underlying to a Dirac-Kähler type equation on the lattice. We make use of the Cayley transform ϕ(w) = 1 + w 1 − w to show that the resulting group representation arise from the same mathematical framework as the conformal group representation in terms of the general linear group GL (2, Γ(n − 1, n − 1) ∪ {0}). That allows us to describe such class of solutions as a commutative n−ary product, involving the quasi-monomials ϕ (z j ) − x j h (x j ∈ hZ) with membership in the paravector space R ⊕ Re j e n+j .

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“…These properties allow to prove that e F (z1,z2) = P(z 1 , z 2 ), where P(z 1 , z 2 ) is the hypercomplex Pascal matrix introduced in [13].…”

Section: Particular Block Matricesmentioning

confidence: 99%

“…Let t = (t 1 , t 2 ) ∈ R 2 , z = (z 1 , z 2 ) ∈ H 2 , and consider a hypercomplex exponential function defined as in [13] by the throughout convergent series…”

Section: Hypercomplex Laguerre Polynomialsmentioning

confidence: 99%

Laguerre Polynomials in Several Hypercomplex Variables and Their Matrix Representation

Malonek

Tomaz

2011

Computational Science and Its Applications - ICCSA 2011

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Abstract. Recently the creation matrix, intimately related to the Pascal matrix and its generalizations, has been used to develop matrix representations of special polynomials, in particular Appell polynomials. In this paper we describe a matrix approach to polynomials in several hypercomplex variables based on special block matrices whose structures simulate the creation matrix and the Pascal matrix. We apply the approach to hypercomplex Laguerre polynomials, although it can be used for other Appell sequences, too.

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Bernoulli polynomials and Pascal matrices in the context of Clifford analysis

Cited by 14 publications

References 13 publications

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

A conformal group approach to the Dirac–Kähler system on the lattice

Laguerre Polynomials in Several Hypercomplex Variables and Their Matrix Representation

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