Abstract
This paper aims to give new insights into homogeneous hypercomplex Appell polynomials through the study of some interesting arithmetical properties of their coefficients. Here Appell polynomials are introduced as constituting a hypercomplex generalized geometric series whose fundamental role sometimes seems to have been neglected. Surprisingly, in the simplest non-commutative case their rational coefficient sequence reduces to a coefficient sequence \({\mathcal {S}}\) used in a celebrated theorem on positive trigonometric sums by Vietoris (Sitzungsber Österr Akad Wiss 167:125–135, 1958). For \({\mathcal {S}}\) a generating function is obtained which allows to derive an interesting relation to a result deduced by Askey and Steinig (Trans AMS 187(1):295–307, 1974) about some trigonometric series. The further study of \({\mathcal {S}}\) is concerned with a sequence of integers leading to its irreducible representation and its relation to central binomial coefficients.
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Notes
Quoted by R. Siegmund-Schultze in: Ausgewahlte Kapitel aus der Funktionenlehre, Teubner, Leipzig, 1988, p. 253, including the re-publication of Weierstrass’ “Zur Funktionentheorie,” published in Mittag-Leffler’s commemorative issue of Acta Mathematica, 45 (1925), pp. 1–0.
This is different from the case of several complex variables where in the expansion of the Cauchy kernel a multiple geometric series as generalization of the ordinary geometric series arises.
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Acknowledgements
The work of the first and third authors was supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e Tecnologia”), within project PEst-OE/MAT/UI4106/2013. The work of the second author was supported by Portuguese funds through the CMAT—Centre of Mathematics and FCT within the Project UID/MAT/00013/2013.
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Communicated by Irene Sabadini.
Dedicated to Frank Sommen on the occasion of his 60th birthday.
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Cação, I., Falcão, M.I. & Malonek, H. Hypercomplex Polynomials, Vietoris’ Rational Numbers and a Related Integer Numbers Sequence. Complex Anal. Oper. Theory 11, 1059–1076 (2017). https://doi.org/10.1007/s11785-017-0649-5
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DOI: https://doi.org/10.1007/s11785-017-0649-5