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

Math Methods in App Sciences

2018

In this paper, we exploit the umbral calculus framework to reformulate the so‐called discrete Cauchy‐Kovalevskaya extension in the scope of hypercomplex variables. The key idea is to consider not only formal power series representation for the underlying solution, but also integral representations for the Chebyshev polynomials of first and second kind by means of its Cauchy principal values. It turns out that the resulting integral representation associated to our toy problem is a space‐time Fourier type inversion formula. Moreover, with the aid of some Laplace transform identities involving the generalized Mittag‐Leffler function, we are able to establish a link with a Cauchy problem of differential‐difference type.

L_{t} = \sum_{k = 1}^{\infty} b_{k} \frac{{(\partial_{t})}^{k}}{k!}, with b_{k} = {[{(L_{t})}^{k} t^{k}]}_{t = 0} .

Such construction looks similar to Constales and De Ridder's work (cf Faustino, Remark 4.1), except that the − i L t operator (

i = \sqrt{- 1}

)–appearing quite often on Dirac's equation (cf ,. subsection 5.…”

Section: Problem Setupmentioning

confidence: 71%

“…As studied in depth by the author in a previous study, 19 the above construction yields as a natural exploitation of an abstract framework involving manipulations of shift-invariant operators that admit formal series expansions in terms of the time derivative t as follows:…”

Section: Problem Setupmentioning

confidence: 99%

A note on the discrete Cauchy‐Kovalevskaya extension

Math Methods in App Sciences

2018

“…2 S n 1 . The other was the construction of a one-to-one mapping between the subspaces Re j e nCj of spin C .n, n/ and the (paravector) subspaces H n 1,n j of the Lorentz pseudo-sphere H n 1,n , by means of the stereographic-like projection (14). That was the main key on the proof of Proposition 2.1.…”

Section: Discussionmentioning

confidence: 99%

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

Math Methods in App Sciences

2017

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 .

“…shows us that D h,α may also be seen as a fractional regularization for the discrete Dirac operators on the lattice h 2 Z n , already considered in the series of papers [16,18,19,20].…”

Section: Discrete Dirac-kähler Vs Discrete Laplacian Let Us Take Nomentioning

confidence: 94%

Relativistic Wave Equations on the Lattice: An Operational Perspective

2019

Trends in Mathematics

Dedicated to Professor Wolfgang Sprößig on occasion of his 70th birthday.Abstract. This paper presents an operational framework for the computation of the discretized solutions for relativistic equations of Klein-Gordon and Dirac type. The proposed method relies on the construction of an evolutiontype operador from the knowledge of the Exponential Generating Function (EGF), carrying a degree lowering operator Lt = L(∂t). We also use certain operational properties of the discrete Fourier transform over the n−dimensional Brioullin zone Q h = − π h , π h n -a toroidal Fourier transform in disguise -to describe the discrete counterparts of the continuum wave propagators,∆ − m 2 respectively, as discrete convolution operators. In this way, a huge class of discretized time-evolution problems of differential-difference and difference-difference type may be studied in the spirit of hypercomplex variables.