The paper is focused on intrinsic properties of a one parameter family of non-symmetric number triangles T(n), n ≥ 2, which
arises in the construction of hyperholomorphic Appell polynomials

Cação, Isabel

Falcão, Irene

Malonek, Helmuth

Tomaz, Graça

Repositório Institucional do Instituto Politécnico da Guarda

Non-symmetric Number Triangles Arisingfrom Hypercomplex Function Theoryin Rn+1Isabel Cação1 , M. Irene Falcão2 , Helmuth R. Malonek1 ,and Graça Tomaz1,3(B)1 CIDMA, Universidade de Aveiro, Aveiro, Portugal{isabel.cacao,hrmalon}@ua.pt2 CMAT, Universidade do Minho, Braga, Portugalmif@math.uminho.pt3 Instituto Politécnico da Guarda, Guarda, Portugalgtomaz@ipg.ptAbstract. The paper is focused on intrinsic properties of a one-parameter family of non-symmetric number triangles T (n), n ≥ 2, whicharises in the construction of hyperholomorphic Appell polynomials.Keywords: Non-symmetric Pascal triangle · Clifford algebra ·Recurrence relation1 IntroductionA one-parameter family of non-symmetric Pascal triangles was considered in[8] and a set of its basic properties was proved. Such family arises from studieson generalized Appell polynomials in the framework of Hypercomplex FunctionTheory in Rn+1, n ≥ 1, (cf. [7]). If n ≥ 2, it is given by the infinite triangulararray, T (n), of rational numbersT ks (n) =(ks)(n+12 )k−s(n−12 )s(n)k, k = 1, 2, . . . , ; s = 0, 1, . . . , k, (1)where (a)r := a(a + 1) . . . (a + r − 1), for any integer r ≥ 1, is the Pochhammersymbol with (a)0 := 1, a ≥ 0. If n = 1, then the triangle degenerates to a uniquecolumn because T k0 (1) ≡ 1 and, as usual T ks (1) := 0, s > 0.The non-symmetric structure of this triangle T (n) is a consequence of thepeculiarities of a non-commutative Clifford algebra C0,n frequently used in prob-lems of higher dimensional Harmonic Analysis, like the solution of spinor sys-tems as n-dimensional generalization of Dirac equations and their application inQuantum Mechanics and Quantum-Field Theory [6].Hypercomplex Function Theory in Rn+1 is a natural generalization of the clas-sical function theory of one complex variable in the framework of Clifford Alge-bras. The case n > 1 extends the complex case to paravector valued functionsc© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022O. Gervasi et al. (Eds.): ICCSA 2022 Workshops, LNCS 13377, pp. 420–434, 2022.https://doi.org/10.1007/978-3-031-10536-4_28

Non-symmetric Number Triangles Arising from Hypercomplex Function Theory in IR^{n+1}

http://bdigital.ipg.pt/dspace/bitstream/10314/6094/1/Artigo-2022.pdf

Non-symmetric Number Triangles Arising from Hypercomplex Function Theory in IR^{n+1}

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