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A note on a one-parameter family of non-symmetric number triangles. (English) Zbl 1268.30041
The authors consider the following infinite array of numbers \[ T_s^k(n) = \frac{k!}{n_{(k)}}\frac{\left(\frac{n+1}{2}\right)_{(k-s)}}{(k-s)!}\frac{\left(\frac{n-1}{2}\right)_{(s)}}{s!}, \quad n,s \in \mathbb{N}, \; s=0,1, \dots, k \] where \(a_{(r)}\) denotes the Pochhammer symbol. These number were introduced to construct generalized Appell polynomials in the framework of Clifford analysis.
In this paper the relation of these numbers with the elements of the Pascal triangle and the coefficients of the geometric series is discussed. Several arithmetical properties of the family of number triangles composed by fractions \(T_s^k(n)\) for different parameter values \(n\) in lines of height \(k\) and ordered from \(s =0\) up to \(s=k\) are proven.

MSC:
30G35 Functions of hypercomplex variables and generalized variables
05A19 Combinatorial identities, bijective combinatorics
11B65 Binomial coefficients; factorials; \(q\)-identities
Software:
Quaternions
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