Abstract Related to generalized arithmetic triangle, we introduce the hyper (r,q)-Fibonacci polynomials as the sum of these elements along a finite ray starting from a specific point, which generalize the hyper-Fibonacci polynomials. We give generating function, recurrence relations and we show some properties whose application allows us to extend the notion of Cassini determinant and to study some ratios. Moreover, we derive a connection between these polynomials and the incomplete (r,q)-Fibonacci polynomials defined in this paper.

ABSTRACT This paper is interested in a new sequence of linear positive operators including degenerate Appell polynomials. We give a convergence theorem for these operators and obtain the quantitative estimation of the approximation by using modulus of continuity, Peetre’s 𝒦-functional, Lipschitz class functions and a Voronovskaja-type theorem. In addition, we give a Kantorovich modification of these operators and derive some approximation properties.

ABSTRACT Let X, Y be two Banach spaces over K = ℝ \[\mathbb{K}=\mathbb{R}\] or ℂ \[\mathbb{C}\] , and let f := F+C be a weakly coercive operator from X onto Y, where F is a Fredholm proper operator, and C is a C 1-compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C 1-diffeomorphism. When one of these conditions does not hold and instead y is a regular value, the equation f(x) = y has at most finite number of solutions. As a consequence of the main result two corollaries are given. A second theorem studies the finite dimensional case. As an application, one example is given. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems.

ABSTRACT In this paper, we characterize the perfect zero-divisor graphs and chordal zero-divisor graphs (its complement) of ordered sets. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graphs of rings, the annihilating ideal graphs of rings, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of groups. In fact, it is shown that these graphs associated with a commutative ring R with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from R.

ABSTRACTLetk≥ 2 be a fixed integer. Thek-generalized Lucas sequence{Ln(k)}n≥0\[{{\left\{ L_{n}^{\left( k \right)} \right\}}_{n\ge 0}}\]starts with the positive integer initial valuesk, 1, 3, …, 2k−1– 1, and each term afterward is the sum of thekconsecutive preceding elements. An integernis said to be close to a positive integermifnsatisfies|n−m|<m\[\left| n-m \right|<\sqrt{m}\]. In this paper, we combine these two concepts. We solve completely the diophantine inequality|Ln(k)−2m|<2m/2\[\left| L_{n}^{\left( k \right)}-{{2}^{m}} \right|<{{2}^{m/2}}\]in the non-negative integersk,n, andm. This problem is equivalent to the resolution of the equationLn(k)=2m+t\[L_{n}^{\left( k \right)}={{2}^{m}}+t\]with the condition |t| < 2m/2,t∈ℤ\[t\in \mathbb{Z}\]. We also discovered a new formula forLn(k)\[L_{n}^{\left( k \right)}\]which was very useful in the investigation of one particular case of the problem.

ABSTRACT Let K be a number field generated by a root θ of a monic irreducible trinomial F ( x ) = x n + a x m + b ∈ ℤ [ x ] . In this paper, we study the problem of monogenity of K. More precisely, we provide some explicit conditions on a, b, n, and m for which K is not monogenic. As applications, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree n = 2 r · 3 k with r and k two positive integers. We also give infinite families of non-monogenic sextic number fields defined by trinomials. Some illustrating examples are giving at the end of this paper.

ABSTRACT In a way different from traditional methods, recently, new series spaces have been established by using absolute summability methods. Among these absolute summability methods, the absolute generalized Nörlund summability method |N, p n , q n , θ n | k is extremely important as it includes several summability methods such as absolute Nörlund, weighted, Cesàro and Euler summability methods. Now, it is natural to ask behavior of the space derived by the absolute summability of generalized Nörlund means. The purpose of this study is to derive a more general series space including some well known spaces such as the spaces | N ˜ q θ | k ${\left| {\tilde N_q^\theta } \right|_k}$ , |C α | k , |C α, β | k , | N p θ | k ${\left| {N_p^\theta } \right|_k}$ and | E r θ | k ${\left| {E_r^\theta } \right|_k}$ , and to investigate its topological structure, α-, β- and γ-duals. Beside this, some matrix transformations related to these spaces have been characterized.

ABSTRACT We show that there is a set of 22 κ ultrafilters incomparable in the Rudin-Frolík order of βκ\κ, where κ is regular, for which no subset with more than one element has an infimum.

ABSTRACT Our goal in this paper is to compute the 2-rank of the class group of the fields K = ℚ ( − 2 , q , d ) \[\mathbb{K}=\mathbb{Q}\left( \sqrt{-2},\sqrt{q},\sqrt{d} \right)\] for any odd positive squarefree integer d and any prime integer q ≡ 5 (mod 8). Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic, of type (2, 2), (2, 4) or (2, 2, 2).

ABSTRACT We prove a lower bound estimate for Hajłasz-Besov capacity in metric spaces in terms of Netrusov-Hausdorff content. We also prove a similar estimate for Hajłasz-Triebel-Lizorkin capacity in terms of Hausdoroff content. These results are improvements of the earlier results obtained by Nuutinen in 2016 and the first author in 2020.