About Pascal's tetrahedron with hypercomplex entries

Abstract

It is evident, that the properties of monogenic polynomials in (n + 1)−real variables significantly depend on the generators e 1 , e 2 ,. .. , e n of the underlying 2 n-dimensional Clifford algebra Cℓ 0,n over R and their interactions under multiplication. The case of n = 3 is studied through the consideration of Pascal's tetrahedron with hypercomplex entries as special case of the general Pascal simplex for arbitrary n, which represents a useful geometric arrangement of all possible products. The different layers L k of Pascal's tetrahedron (or pyramid) are built by ordered symmetric products contained in the trinomial expansion of (e 1 + e 2 + e 3) k , k = 0, 1,. .. .

A Sturm‐Liouville equation on the crossroads of continuous and discrete hypercomplex analysis

“…As a first step, the case n = 3 and the corresponding hypercomplex Pascal polyhedron has been studied. 52 A remark on the expression of complicated formulae in representation theory and hypergeometric functions theory by polyhedral geometry can be found in Arnolds paper. 1 Due to the fact that the inner structure of all  n k is in a certain sense independent from n (only the dimension of R n , i.e.…”

“…The higher dimensional case with n ≥ 3 needs to be handled with polyhedral geometry. As a first step, the case and the corresponding hypercomplex Pascal polyhedron has been studied 52 . A remark on the expression of complicated formulae in representation theory and hypergeometric functions theory by polyhedral geometry can be found in Arnolds paper 1 …”

Preface: International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2017)

“…In the following we intensively use the embedding of the non-commutative Clifford algebra product into an n − nary symmetric product (see [4] and more detailed [5]):…”

Combinatorial identities in the context of hypercomplex function theory

“…In the following we intensively use the embedding of the non-commutative Clifford algebra product into an n − nary symmetric product (see [4] and more detailed [5]):…”

“…(2), if k is even(8) which, in turn, can be written in the form c k (2) in terms of the generalized central binomial coefficient resp. in terms of Pochhammer symbols as in(5). Obviously, (8) is the special n = 2 case of c k (n) 1)!!…”