Clifford Analysis between Continuous and Discrete

Abstract

Some decades ago D. Knuth et al. have coined concrete mathematics as the blending of CONtinuous and disCRETE math, taking into account that problems of standard discrete mathematics can often be solved by methods based on continuous mathematics together with a controlled manipulation of mathematical formulas. Of course, it was not a new idea, but due to the ongoing emergence of computer aided algebraic manipulation tools of that time it emphasized their use for elegant solutions of old problems or even the detection of new important relationships. Our aim is to show that the same philosophy can be successfully applied to Clifford Analysis by taking advantages of its inherent non-commutative algebra to obtain results or develop methods that are different from other ones. In particular, we determine new binomial sums by using a hypercomplex generating function for a special type of monogenic polynomials and develop an algorithm for the determination of their scalar and vector part which illustrates well the differences to the corresponding complex case.

(Discrete) Almansi type decompositions: an umbral calculus framework based on osp(1|2) symmetries

“…Roughly speaking, the construction of hypercomplex Bernoulli polynomials shall be obtained via Appell sets [8,34]. This was investigated at an early stage by Abul-ez & Constales in [1] in terms of basic sets of hypercomplex polynomials.…”

“…[22]) that families of discrete polynomials can be constructed as a blending between continuous and discrete Clifford analysis giving an affirmative answer to the paper of Malonek & Falcão (cf. [34]).…”

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

“…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.…”

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

“…Several approaches for finding systems of solutions associated to discretized versions of (1) through combinatorics (cf. [MF08,MT08]), Lie-algebraic representations (cf. [RSKS10,FR11,F13]), and a combination of both (cf.…”