Computer assisted solution of systems of two variable linear functional equations

Borus, Gergő Gyula; Gilányi, Attila

doi:10.1007/s00010-020-00736-z

Cited by 9 publications

(3 citation statements)

References 24 publications

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“…, f n+1 solve (1). Then Theorem 1 implies that each of the functions f i is of the form (7), thus, it is a polynomial function of degree n. Therefore, each of the functions ϕ i is a decent solution of the polynomial congruence of degree n. Now, assume that ϕ i = f i +g i for i = 0, . .…”

Section: Resultsmentioning

confidence: 99%

“…Its solutions, in a general case, were determined by Székelyhidi [22,23]. A computer program presenting the solutions of functional equations of type (1) was described in [7] (cf., also, [5,6,10,11]). Problems connected to class (1), its generalizations and its applications have been studied by several authors during the last more than 100 years.…”

Section: Introductionmentioning

confidence: 99%

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On linear functional equations modulo $${\mathbb {Z}}$$

Gilányi

Lewicka

2021

Aequat. Math.

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In this paper, we consider the condition $$\sum _{i=0}^{n+1}\varphi _i(r_ix+q_iy)\in {\mathbb {Z}}$$ ∑ i = 0 n + 1 φ i ( r i x + q i y ) ∈ Z for real valued functions defined on a linear space V. We derive necessary and sufficient conditions for functions satisfying this condition to be decent in the following sense: there exist functions $$f_i:V\rightarrow {\mathbb {R}}$$ f i : V → R , $$g_i:V\rightarrow {\mathbb {Z}}$$ g i : V → Z such that $$\varphi _i=f_i+g_i$$ φ i = f i + g i , $$(i=0,\dots ,n+1)$$ ( i = 0 , ⋯ , n + 1 ) and $$\sum _{i=0}^{n+1}f_i(r_ix+q_iy)=0$$ ∑ i = 0 n + 1 f i ( r i x + q i y ) = 0 for all $$x, y\in V$$ x , y ∈ V .

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On linear functional equations modulo $${\mathbb {Z}}$$

Gilányi

Lewicka

2021

Aequat. Math.

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“…Instead, we may formulate results which are valid for large classes of equations. It is even possible to write computer programs which solve linear functional equations, see the papers of Gilányi [13] and Borus and Gilányi [5].…”

Section: Introductionmentioning

confidence: 99%

On a new class of functional equations satisfied by polynomial functions

et al. 2021

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The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi’s result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation $$\begin{aligned} F(x + y) - F(x) - F(y) = yf(x) + xf(y) \end{aligned}$$ F ( x + y ) - F ( x ) - F ( y ) = y f ( x ) + x f ( y ) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.

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