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On n-associative formal power series

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Abstract

A formal power series \({F(x_{1}, \ldots, x_{n})\in\mathbb{C}[\![x_1,\ldots x_n]\!]}\) of order at least 1 is called n-associative, n ≥ 3, if

$$F(F(x_{1}, \ldots, x_{n}), x_{n+1},\ldots,x_{2n-1})=\cdots= F(x_1,\ldots,x_{n-1},F(x_n,x_{n+1},\ldots,x_{2n-1})).$$

This notion generalizes associativity which is the special case of n = 2. We determine the set of all n-associative formal power series over \({\mathbb{C}}\), all convergent n-associative power series, and all commutative (or symmetric) n-associative formal power series. Moreover we study relations between n- and m-associativity for certain \({n,m\in\mathbb{N}}\) and determine the structure of associative families (F n ) n ≥ 1 of formal power series \({F(x_{1}, \ldots, x_{n})\in\mathbb{C}[\![x_1,\ldots x_n]\!]}\).

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  1. Institut für Mathematik und Wissenschaftliches Rechnen, NAWI-Graz Universitä Graz, Heinrichstr. 36/4, 8010, Graz, Austria

    Harald Fripertinger

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  1. Harald Fripertinger
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Correspondence to Harald Fripertinger.

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Fripertinger, H. On n-associative formal power series. Aequat. Math. 90, 449–467 (2016). https://doi.org/10.1007/s00010-015-0372-0

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  • DOI: https://doi.org/10.1007/s00010-015-0372-0

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