Authors:Bilender Allahverdiev, Huseyin Tuna Pages: 1 - 15 Abstract: In this paper, a singular linear \(q\)-Hamiltonian system is considered. The Titchmarsh-Weyl theory for this problem is constructed. Firstly, we provide some necessary fundamental concepts of the \(q\)-calculus. Later, we studied Titchmarsh-Weyl functions \(M\left( \lambda\right)\) and circles \(\mathcal{C}_{TW}\left(a,\lambda\right)\) for this system. Circles \(\mathcal{C}_{TW}\left(a,\lambda\right)\) are proved to be nested. In the fourth part of the work, the number of square-integrable solutions of this system is studied. In the fifth part of the work, boundary conditions in the singular case are obtained. Finally, a self-adjoint operator is defined. PubDate: 2024-07-29 DOI: 10.17951/a.2024.78.1.1-15 Issue No:Vol. 78, No. 1 (2024)
Authors:Athanasios Beslikas Pages: 17 - 26 Abstract: In this note we provide Holland-Walsh-type characterizations for functions on the \(\mathcal{N}(p,q,s)\) spaces on the unit ball for specific values of \(p\ge 1\). Characterizations for the holomorphic function spaces \(\mathcal{N}(p,q,s)\) were studied extensively by B. Hu and S. Li. PubDate: 2024-07-29 DOI: 10.17951/a.2024.78.1.17-26 Issue No:Vol. 78, No. 1 (2024)
Authors:Istvan Fazekas, Nyanga Honda Masasila Pages: 27 - 35 Abstract: A general tool to prove conditional strong laws of larger number is considered. It is shown that a conditional Kolmogorov type inequality implies a conditional Hajek–Renyi type inequality and this implies a strong law of large numbers. Both probability and moment inequalities are considered. Some applications are offered in the last section. PubDate: 2024-07-29 DOI: 10.17951/a.2024.78.1.27-35 Issue No:Vol. 78, No. 1 (2024)
Authors:Per Karlsson, Thomas Ernst Pages: 37 - 73 Abstract: The purpose of this paper is to consider five classes of quadratic and cubic hypergeometric transformations in the spirit of Bailey and Whipple. We shall successfully evaluate several hypergeometric functions, of the types \(_{2}\text{F}_{1}(x)\), \(_{3}\text{F}_{2}(x)\), and \(_{4}\text{F}_{3}(x)\), with each function having one or more free parameters, and with the argument $x$ chosen to equal such unusual values as \(x=\pm 1,-8,\frac 14, -\frac 18\), (these four values having been explored initially by Gessel and Stanton). In each case, companion identities and/or inverse transformations are given, which are sometimes proved by a limiting process for a divergent hypergeometric series. Some of the proofs use the Clausen quadratic formula, Euler reflection formula, Legendre duplication, Gauss multiplication formula, Euler transformation, hypergeometric reversion formula and known hypergeometric summation formulas. The proofs in the terminating case are simpler and can lead to mixed summation formulas, which depend on values of a negative integer. Some of the formulas use the Digamma function and a dimension formula is referred to. PubDate: 2024-07-29 DOI: 10.17951/a.2024.78.1.37-73 Issue No:Vol. 78, No. 1 (2024)
Authors:Jan Kurek, Włodzimierz Mikulski Pages: 75 - 86 Abstract: Let \(\mathcal{FM}_{m,n}\) denote the category of fibered manifolds with $m$-dimensional bases and \(n\)-dimensional fibres and their fibered diffeomorphisms onto open images. We describe all \(\mathcal{FM}_{m,n}\)-natural operators \(C\) transforming tuples \((\lambda,g)\) of Lagrangians \(\lambda:J^sY\to\bigwedge ^mT^*M\) (or formal Lagrangians \(\lambda:J^sY\to V^*J^sY\otimes\bigwedge ^mT^*M\)) on \(\mathcal{FM}_{m,n}\)-objects \(Y\to M\) and functions \(g:M\to\mathbf{R}\) into Euler maps \(C(\lambda,g):J^{2s}Y\to V^*Y\otimes\bigwedge^m T^*M\) on \(Y\). The most important example of such \(C\) is the Euler operator \(E\) (from the variational calculus) (or the formal Euler operator \(\mathbf{E}\)) treated as the operator in question depending only on Lagrangians (or formal Lagrangians). PubDate: 2024-07-29 DOI: 10.17951/a.2024.78.1.75-86 Issue No:Vol. 78, No. 1 (2024)
Authors:Mariola Rubajczyk, Anetta Szynal-Liana Pages: 87 - 95 Abstract: Hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper, we define and study hybrid numbers with cobalancing and Lucas-cobalancing coefficients. We derive some fundamental identities for these numbers, among others the Binet formulas and the general bilinear index-reduction formulas which imply the Catalan, Cassini, Vajda, d’Ocagne and Halton identities. Moreover, the generating functions for cobalancing and Lucas-cobalancing hybrid numbers are presented. PubDate: 2024-07-29 DOI: 10.17951/a.2024.78.1.87-95 Issue No:Vol. 78, No. 1 (2024)
Open Access journal
[26 journals]
