The vertex-edge Wiener polynomials of a simple connected graph are defined based on the distances between vertices and edges of that graph. The first derivative of these polynomials at one are called the vertex-edge Wiener indices. In this paper, we express some basic properties of the first and second vertex-edge Wiener polynomials of simple connected graphs and compare the first and second ve...

We show that the self-shuffle of Thue-Morse given by Charlier et al. is optimal/canonical in the sense that among self-shuffles of Thue-Morse, it has the lexicographically least directive sequence starting with 1.

Let μ (q) p be the q-deformed Poisson measure in the sense of SaitohYoshida [24] and νp be the measure given by Equation (3.6). In this short paper, we introduce the q-deformed analogue of the Segal-Bargmann transform associated with μ (q) p . We prove that our Segal-Bargmann transform is a unitary map of L(μ (q) p ) onto the q-deformed Hardy spaceH (νq). Moreover, we give the Segal-Bargmann re...

Hermite-Gauss and Laguerre-Gauss modes of a continuous optical field in two dimensions can be obtained from each other through paraxial optical setups that produce rotations in (four-dimensional) phase space. These transformations build the SU(2) Fourier group that is represented by rigid rotations of the Poincaré sphere. In finite systems, where the emitters and the sensors are in NxN square p...

In this paper we establish several polynomials similar to Bernstein's polynomials and several refinements of  Hermite-Hadamard inequality for convex functions.

Volodymyr P. Kravchuk, 2, ∗ Denis D. Sheka, Attila Kákay, Oleksii M. Volkov, Ulrich K. Rößler, Jeroen van den Brink, 5 Denys Makarov, and Yuri Gaididei Bogolyubov Institute for Theoretical Physics of National Academy of Sciences of Ukraine, 03680 Kyiv, Ukraine Leibniz-Institut für Festkörperund Werkstoffforschung, IFW Dresden, D-01171 Dresden, Germany Taras Shevchenko National University of Kyi...

We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble is related to a two-dimensional directed growth model, and the Charlier ensemble is related to the...

We show that the function h(x) = ∏ i<j(xj − xi) is harmonic for any random walk in Rk with exchangeable increments, provided the required moments exist. For the subclass of random walks which can only exit the Weyl chamber W = {x : x1 < x2 < · · · < xk} onto a point where h vanishes, we define the corresponding Doob h-transform. For certain special cases, we show that the marginal distribution ...