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In this paper, a computer program developed in the computer algebra system Maple is presented, which investigates alienness and strong alienness of linear functional equations. [ABSTRACT FROM AUTHOR]

In this paper, we utilize some known Steffensen-type identities, obtained by using Lidstone's interpolating polynomial, to prove new generalizations of Steffensen's inequality. We obtain these new generalizations by using the weighted Hermite-Hadamard inequality for (2 n + 2) - convex and (2 n + 3) - convex functions. Further, the newly obtained inequalities can be observed as an upper- and lower-bound for utilized Steffensen-type identities. [ABSTRACT FROM AUTHOR]

In this paper, we consider the affine variety codes obtained evaluating the polynomials b y = a k x k + ... + a 1 x + a 0 , b , a i ∈ F q r , at the affine F q r -rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolute irreducibility of certain algebraic varieties. [ABSTRACT FROM AUTHOR]

In the present paper, a general class of linear functional equations is considered and a computer program is described, which determines the exact solutions of systems of equations belonging to this class. [ABSTRACT FROM AUTHOR]

In this paper we consider generalized monomial functions f , g : F → C (of possibly different degree) that also fulfill f (P (x)) = Q (g (x)) x ∈ F , where P ∈ F [ x ] and Q ∈ C [ x ] are given (classical) polynomials. [ABSTRACT FROM AUTHOR]

The aim of this paper is to fill in the gaps in the formulation and the proof of a theorem contained in the paper by K.Ozeki (Aequ Math 25:247–252, 1982) published in this journal. We also give a short proof of this theorem and use it to obtain certain information about the factorization of polynomials of the form f (x) - f (y) . [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

Spectral analysis and synthesis studies translation invariant function spaces, so-called varieties over topological groups. The basic building blocks are the finite dimensional varieties. In the commutative case finite dimensional varieties are spanned by exponential polynomials. In non-commutative situations no relevant results exist. In this paper we consider finite dimensional left translation invariant linear spaces of continuous complex valued functions over the Heisenberg group. Using basic knowledge about Lie algebra we describe all left varieties of this type. In particular, it turns out that those function spaces are spanned by exponential polynomials as well. [ABSTRACT FROM AUTHOR]

In this paper we consider a generalized monomial or polynomial f : R → R that satisfies the additional equation f (x) f (y) = 0 for the pairs (x , y) ∈ D , where D ⊆ R 2 is given by some algebraic condition. In the particular cases when f is a generalized polynomial and there exist non-constant regular polynomials p and q that fulfill D = { (p (t) , q (t)) | t ∈ R } or f is a generalized monomial and there exists a positive rational m fulfilling D = { (x , y) ∈ R 2 | x 2 - m y 2 = 1 } , we prove that f (x) = 0 for all x ∈ R . [ABSTRACT FROM AUTHOR]

In order to discuss the complexity of nonmonotonicity of a given PM function, it is needed to determine the height of PM functions. But it is not easy to do even for a polynomial because there are great difficulties in determining the number of real zeros for polynomials of higher degrees. In this paper we introduce two iterative sets to determine the nonmonotonicity height for polynomials and give a numerical algorithm for computing the height of general polynomials. [ABSTRACT FROM AUTHOR]

The study of quantum integers and their operations is closely related to the studies of symmetries of roots of polynomials and of fundamental questions of decompositions in Additive Number Theory. In his papers on quantum arithmetics, Melvyn Nathanson raises the question of classifying solutions of functional equations arising from the multiplication of quantum integers, starting with polynomial solutions and then generalizing to rational function solutions. The classification of polynomial solutions with fields of coefficients of characteristic zero and support base P has been completed. In a paper concerning the Grothendieck group associated to the collection of polynomial solutions, Nathanson poses a problem which asks whether the set of rational function solutions strictly contains the set of ratios of polynomial solutions. It is now known that there are infinitely many rational function solutions Γ with fields of coefficients of characteristic zero not constructible as ratios of polynomial solutions, even in the purely cyclotomic case, which is the case most similar to the polynomial solution case. The classification of polynomial solutions is thus not sufficient, in essential ways, to resolve the classification problem of all rational function solutions with fields of coefficients of characteristic zero. In this paper we study symmetries of roots of rational functions and the classification of the important class-the last and main obstruction to the classification problem-of rational function solutions, the purely cyclotomic, purely nonrational primitive solutions with fields of coefficients of characteristic zero and support base P, which allows us to complete the classification problem raised by Nathanson. [ABSTRACT FROM AUTHOR]

We examine the family of all (at most) three-point symmetric quadratures on [ - 1 , 1 ] which are exact on polynomials of order 3 to find all possible inequalities between them in the class of 3-convex functions. Next we optimise them by using convex combinations of the quadratures considered. We find the optimal quadrature and use it to construct the adaptive method of approximate integration. An effective method to estimate the error of this method is also given. It needs a considerably fewer number of subdivisions of the interval of integration than the classical adaptive methods as well as the method developed by the second-named author in his recent paper (Wa̧sowicz in Aequ Math 94(5):887–898, 2020). [ABSTRACT FROM AUTHOR]

The aim of the present paper is to describe some properties of functions with finite dimensional difference spaces by means of spectral analysis and spectral synthesis. We are going to apply these results to a version of a Levi-Civita functional equation, which has been recently studied by J. M. Almira and E. Shulman. [ABSTRACT FROM AUTHOR]

In 1970, Coxeter gave a short and elegant geometric proof showing that if p1,p2,…,pn are vertices of an n-gon P in cyclic order, then P is affinely regular if, and only if there is some λ≥0 such that pj+2-pj-1=λ(pj+1-pj) for j=1,2,…,n. The aim of this paper is to examine the properties of polygons whose vertices p1,p2,…,pn∈C satisfy the property that pj+m1-pj+m2=w(pj+k-pj) for some w∈C and m1,m2,k∈Z. In particular, we show that in ‘most’ cases this implies that the polygon is affinely regular, but in some special cases there are polygons which satisfy this property but are not affinely regular. The proofs are based on the use of linear algebraic and number theoretic tools. In addition, we apply our method to characterize polytopes with certain symmetry groups. [ABSTRACT FROM AUTHOR]

After his extensive study of Whitney numbers, Benoumhani introduced Dowling numbers and polynomials as generalizations of the well-known Bell numbers and polynomials. Later, Cheon and Jung gave the r-generalization of these notions. Based on our recent combinatorial interpretation of r-Whitney numbers, in this paper we derive several new properties of r-Dowling polynomials and we present alternative proofs of some previously known ones. [ABSTRACT FROM AUTHOR]

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials {xN}N=0∞. We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials. [ABSTRACT FROM AUTHOR]

The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of all bivariate polynomials that satisfy the Jacobi identity over infinite integral domains. Although this description depends on the characteristic of the domain, it turns out that all these polynomials are of degree at most one in each indeterminate. [ABSTRACT FROM AUTHOR]

We study the stability properties of the equation which is motivated by numerical integration. In Szostok and Wa̧sowicz (Appl Math Lett 24(4):541-544, ) the stability of the simplest equation of the type (0.1) was investigated thus the inequality was studied. In the current paper we present a somewhat different approach to the problem of stability of (0.1). Namely, we deal with the inequality [ABSTRACT FROM AUTHOR]

In this paper local polynomials on Abelian groups are characterized by a 'local' Fréchet-type functional equation. We apply our result to generalize Montel's Theorem and to obtain Montel-type theorems on commutative groups. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to determine the general (and also the continuous) solutions of the discrete wave equation, that is, to solve the following partial difference equation [ABSTRACT FROM AUTHOR]

Given rational vector spaces V, W a mapping $${f \colon V \to W}$$ is called a generalized polynomial of degree at most n, if there are homogeneous generalized polynomials f of degree i such that $${f = \sum_{i = 0}^n f_i}$$ . Homogeneous generalized polynomials f of degree i are mappings of the form $${f_i (x) = f_i^*(x, x, \ldots , x)}$$ with $${f_i^* \colon V^i \to W i}$$ -linear. In the literature one may find quite a lot of functional equations such that their general solution is of the form f or $${f_n + f_{n - 1}}$$ where n is a small positive integer ( ≤ 6 or ≤ 4 respectively). In this paper, given an arbitrary positive integer n and an arbitrary subset $${L \subseteq \{0, 1, \ldots, n\}}$$ such that $${n \in L}$$ , a method is described to find (many) functional equations, such that their general solution is given by $${\sum_{i \in L} f_i}$$ . For the cases $${L = \{n\}}$$ and $${L = \{n - 1, n\}}$$ additional equations are given. [ABSTRACT FROM AUTHOR]

In this paper we prove that if a generalized polynomial function f satisfies the condition f( x) f( y) = 0 for all solutions of the equation x + y = 1, then f is identically equal to 0. [ABSTRACT FROM AUTHOR]

Let G be an Abelian group, and let C G denote the set of complex valued functions defined on G. A map D : C G → C G is a difference operator, if there are complex numbers a i and elements b i ∈ G (i = 1 , ... , n) such that (D f) (x) = ∑ i = 1 n a i f (x + b i) for every f ∈ C G and x ∈ G . By a system of difference equations we mean a set of equations { D i f = g i : i ∈ I } , where I is an arbitrary set of indices, D i is a difference operator and g i ∈ C G is a given function for every i ∈ I , and f is the unknown function. The solvability cardinal sc (F) of a class of functions F ⊂ C G is the smallest cardinal number κ with the following property: whenever S is a system of difference equations on G such that each subsystem of S of cardinality < κ has a solution in F , then S itself has a solution in F . The behaviour of sc (F) is rather erratic, even for classes of functions defined on R . For example, sc (C [ x ]) = 3 , but sc (TP) = ω 1 , where TP is the set of trigonometric polynomials; sc (C R) = ω , but sc (DF) = (2 ω) + , where DF is the set of functions having the Darboux property. Our aim is to determine or to estimate the solvability cardinal of the class of polynomials defined on R n , on normed linear spaces and, in general, on topological Abelian groups. Let P G denote the class of polynomials defined on the group G. After presenting some general estimates we prove that sc (C [ x 1 , ... , x n ]) = ω if 2 ≤ n < ∞ , and sc (P X) = ω 1 if X is a normed linear space of infinite dimension. For discrete Abelian groups we show that sc (P G) = 3 if r 0 (G) ≤ 1 , sc (P G) = ω if 2 ≤ r 0 (G) < ∞ , and sc (P G) ≥ ω 1 if r 0 (G) is infinite, where r 0 (G) denotes the torsion free rank of G. The solvability of systems of difference equations is closely connected to the existence of projections of function classes commuting with translations (see Theorem 7.1). As an application we construct a projection from C R n onto C [ x 1 , ... , x n ] commuting with translations by vectors having rational coordinates (Theorem 7.4). [ABSTRACT FROM AUTHOR]