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See detailOn indefinite sums weighted by periodic sequences
Marichal, Jean-Luc UL

in Results in Mathematics (2019), 74(3), 95

For any integer $q\geq 2$ we provide a formula to express indefinite sums of a sequence $(f(n))_{n\geq 0}$ weighted by $q$-periodic sequences in terms of indefinite sums of sequences $(f(qn+p))_{n\geq 0 ... [more ▼]

For any integer $q\geq 2$ we provide a formula to express indefinite sums of a sequence $(f(n))_{n\geq 0}$ weighted by $q$-periodic sequences in terms of indefinite sums of sequences $(f(qn+p))_{n\geq 0}$, where $p\in\{0,\ldots,q-1\}$. When explicit expressions for the latter sums are available, this formula immediately provides explicit expressions for the former sums. We also illustrate this formula through some examples. [less ▲]

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See detailA new invariance identity and means
Devillet, Jimmy UL; Matkowski, Janusz

in Results in Mathematics (2018), 73(4), 130

The invariance identity involving three operations D_{f,g} : X x X -> X of the form D_{f,g} (x,y) = (f o g)^{-1} (f (x) + g (y)) , is proposed. The connections of these operations with means is ... [more ▼]

The invariance identity involving three operations D_{f,g} : X x X -> X of the form D_{f,g} (x,y) = (f o g)^{-1} (f (x) + g (y)) , is proposed. The connections of these operations with means is investigated. The question when the invariance equality admits three means leads to a com- posite functional equation. Problem to determine its continuous solutions is posed [less ▲]

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See detailOn functional equations characterizing derivations: methods and examples
Gselmann, Eszter; Kiss, Gergely UL; Vincze, Csaba

in Results in Mathematics (2018)

Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions ... [more ▼]

Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive functions satisfying some further functional equations as well. It is an important question that how these morphisms can be characterized among additive mappings in general. The paper contains some multivariate characterizations of higher order derivations. The univariate characterizations are given as consequences by the diagonalization of the multivariate formulas. This method allows us to refine the process of computing the solutions of univariate functional equations of the form ∑k=1nxpkfk(xqk)=0, where pk and qk (k=1,…,n) are given nonnegative integers and the unknown functions f1,…,fn:R→R are supposed to be additive on the ring R. It is illustrated by some explicit examples too. As another application of the multivariate setting we use spectral analysis and spectral synthesis in the space of the additive solutions to prove that it is spanned by differential operators. The results are uniformly based on the investigation of the multivariate version of the functional equations. [less ▲]

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See detailA characterization of barycentrically preassociative functions
Marichal, Jean-Luc UL; Teheux, Bruno UL

in Results in Mathematics (2016), 69(1), 245-256

We provide a characterization of the variadic functions which are barycentrically preassociative as compositions of length-preserving associative string functions with one-to-one unary maps. We also ... [more ▼]

We provide a characterization of the variadic functions which are barycentrically preassociative as compositions of length-preserving associative string functions with one-to-one unary maps. We also discuss some consequences of this characterization. [less ▲]

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