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On generalized Vietoris’ number sequences. (English) Zbl 1447.11042
Summary: Recently, by using methods of hypercomplex function theory, the authors [Complex Anal. Oper. Theory 11, No. 5, 1059–1076 (2017; Zbl 1375.30068)] have shown that a certain sequence \(\mathcal{S}\) of rational numbers (Vietoris’ sequence) combines seemingly disperse subjects in real, complex and hypercomplex analysis. This sequence appeared for the first time in a theorem by L. Vietoris [Österr. Akad. Wiss., Math.-Naturw. Kl., S.-Ber., Abt. II 167, 125–135 (1958; Zbl 0088.27402)] with important applications in harmonic analysis [R. Askey and J. Steinig, Trans. Am. Math. Soc. 187, 295–307 (1974; Zbl 0244.42002)] and in the theory of stable holomorphic functions [S. Ruscheweyh and L. Salinas, J. Math. Anal. Appl. 291, No. 2, 596–604 (2004; Zbl 1052.30011)].
A non-standard application of Clifford algebra tools for defining Clifford-holomorphic sequences of Appell polynomials was the hypercomplex context in which a one-parametric generalization \(\mathcal{S}(n)\), \(n \geq 1\), of \(\mathcal{S}\) (corresponding to \(n = 2)\) surprisingly showed up. Without relying on hypercomplex methods this paper demonstrates how purely real methods also lead to \(\mathcal{S}(n)\). For arbitrary \(n \geq 1\) the generating function is determined and for \(n = 2\) a particular case of a recurrence relation similar to that known for Catalan numbers is proved.
MSC:
11B83 Special sequences and polynomials
05A15 Exact enumeration problems, generating functions
11B37 Recurrences
Software:
Quaternions
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References:
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