PROCEEDINGS BOOK OF MICOPAM 2018 Pascal Trapezoids Emerging from Hypercomplex Polynomial Sequences Isabel Ca¸ c˜ ao 1 , M. Irene Falc˜ ao 2 , Helmuth R. Malonek 3 , Gra¸ca Tomaz 3 Abstract The construction of two different representations of special Appell polynomi- als in (n +1) real variables with values in a Clifford algebra suggested to explore the relation between the respective coefficients. Properties of sequences result- ing from such relation and an interesting trapezoidal array of their elements are pointed out. 2010 Mathematics Subject Classifications : 11B83, 05A10, 30G35 Keywords: special sequences, binomial coefficients, Pascal trapezoids, hyper- complex polynomials Introduction In this paper we focus on polynomial sequences in (n +1) real variables with values in the real vector space of paravectors in the corresponding Clifford algebra C ℓ 0,n . We start by introducing some basics of that algebra. The reader can find more details in [4]. Let {e 1 ,e 2 , ··· ,e n } be an orthonormal basis of the real Euclidean vector space R n endowed with a product according to the multiplication rules e i e j + e j e i = −2δ ij , i,j =1, ··· , n, where δ ij is the Kronecker symbol. This non-commutative product generates the associative 2 n −dimensional Clifford algebra C ℓ 0,n over R. The elements z of C ℓ 0,n , called hypercomplex numbers, are of the form z = ∑ A z A e A , where z A ∈ R and the basis {e A : A ⊆{1, ··· ,n}} is formed by e A = e h1 e h2 ··· e hr ,1 ≤ h 1 < ··· <h r ≤ n, e ∅ = e 0 = 1. The vector space R n+1 is embedded in C ℓ 0,n by the identification of the real (n + 1)−tuple (x 0 ,x 1 , ··· ,x n ) with the paravector x = x 0 + x = x 0 + x 1 e 1 + ··· + x n e n ∈A n := span R {1,e 1 ,...,e n }⊂C ℓ 0,n . The conjugate of x ∈A n is given by ¯ x = x 0 − x . The so-called scalar part x 0 and the vector part x of x can be written in the form x 0 =(x +¯ x)/2 and x = (x − ¯ x)/2, respectively. The norm of x is given by |x| =(x ¯ x) 1 2 =(x 2 0 + x 2 1 + ··· + x 2 n ) 1 2 . Consequently, the inverse of each non-zero x is x −1 =¯ x|x| −2 . We consider C ℓ 0,n -valued functions defined as mappings f :Ω ⊂ R n+1 ∼ = A n −→ C ℓ 0,n such that f (x)= ∑ A f A (x)e A ,f A (x) ∈ R and Ω is an open subset of R n+1 ,n ≥ 1. Dedicated to Professor G. Milovanovi´ c 136 Antalya-TURKEY