Shifted Generalized Pascal Matrices in the Context of Clifford Algebra-Valued Polynomial Sequences I.Ca¸c˜ ao 1 , H. R. Malonek 1 , and G. Tomaz 1,2 1 Centro de Investiga¸ c˜ao e Desenvolvimento em Matem´ atica e Aplica¸ c˜ oes, ⋆ Universidade de Aveiro, 3810-193 Aveiro, Portugal 2 Unidade de Investiga¸ c˜ ao para o Desenvolvimento do Interior, Instituto Polit´ ecnico da Guarda, 6300-559 Guarda, Portugal {isabel.cacao, hrmalon}@ua.pt, gtomaz@ipg.pt Abstract. The paper shows the role of shifted generalized Pascal ma- trices in a matrix representation of hypercomplex orthogonal Appell sys- tems. It extends results obtained in previous works in the context of Appell sequences whose first term is a real constant to sequences whose initial term is a suitable chosen polynomial of n variables. Keywords: Shifted generalized Pascal matrix ·Generalized Appell poly- nomials ·Matrix representation 1 Introduction The role of the so-called creation matrix H defined by (H) il =  i, i = l +1 0, i = l +1, i, l =0, 1,...,m, has been highlighted in [2] as the main tool for the matrix representation of Appell polynomial sequences of one real variable and its extension to the rep- resentation of Sheffer sequences (cf. [1]). The importance of the creation matrix was also confirmed in [3] where the authors developed the matrix representation of homogeneous Appell polynomials that are solutions of a generalized Cauchy- Riemann system in Euclidean spaces of arbitrary dimensions. In that work the first term of the considered sequence is a real constant. The consideration of a suitable polynomial in R n as first term leads to more general Appell polynomi- als (see [16]). Those polynomials also appear in the approach used to construct Gelfand-Tsetlin bases related to branching techniques ([7], [15]). The construc- tion process gives relevance to polynomials obtained by shifting coefficients of the ⋆ This work was supported in part by the Portuguese Foundation for Science and Technology (“FCT-Funda¸ c˜ ao para a Ciˆ encia e Tecnologia”), through CIDMA-Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.