Related papers: Convolution numbers: the cyclic case
We retrace the recent history of the Umbral Calculus. After studying the classic results concerning polynomial sequences of binomial type, we generalize to a certain type of logarithmic series. Finally, we demonstrate numerous typical…
We give a characterization of smooth, rotation and dually epi-translation invariant valuations and use this result to obtain a new proof of the Hadwiger theorem on convex functions. We also give a description of the construction of the…
Following Benjamin et al., a matrix with entries being sums of two neighbouring Catalan numbers is considered. Its LU-decomposition is given, by guessing the results and later prove it by computer algebra, with lots of human help.…
A new kind of numbers called Hyper Space Complex Numbers and its algebras are defined and proved. It is with good properties as the classic Complex Numbers, such as expressed in coordinates, triangular and exponent forms and following the…
Nested sums containing binomial coefficients occur in the computation of massive operator matrix elements. Their associated iterated integrals lead to alphabets including radicals, for which we determined a suitable basis. We discuss…
In this paper, we explore a variety of series involving the central binomial coefficients, highlighting their structural properties and connections to other mathematical objects. Specifically, we derive new closed-form representations and…
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
Combinatorial aspects of multivariate diagonal invariants of the symmetric group are studied. As a consequence it is proved the existence of a multivariate extension of the classical Robinson-Schensted correspondence. Further byproduct are…
These notes describe some links between the group $\mathrm{SL}_2(\mathbb{R})$, the Heisenberg group and hypercomplex numbers---complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this…
A class of one-dimensional convolutional codes will be presented. They are all MDS codes, i. e., have the largest distance among all one-dimensional codes of the same length n and overall constraint length delta. Furthermore, their extended…
At the first part of the paper we show how specific umbral extensions of the Stirling numbers of the second kind result in new type of Dobinski-like formulas. In the second part among others one recovers how and why Ward solution of…
Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on nested sums and can be…
A new class of probability distributions closely connected to generalized hyperbolic distributions is introduced. It is more adapted to study the distributions of sums of random number of random variables. The properties of these…
Extremal elements and a h-hull of sets in the n-dimensional hypercomplex space are investigated. Introduced a class of H-quasiconvex sets including strongly hypercomplex convex sets and being closed with respect to intersections.
Three types of integral representations for the cumulative distribution functions of convolutions of non-central p-variate gamma distributions are given by integration of elementary complex functions over the p-cube Cp =…
Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…
For a prime number p, we construct a generating set for the ring of invariants for the p+1 dimensional indecomposable modular representation of a cyclic group of order p^2. We then use the constructed invariants to describe the…
The operational calculus associated with Hermite numbers has been shown to be an effective tool for simplifying the study of special functions. Within this context, Hermite polynomials have been viewed as Newton binomials, with the…
Let $T$ be a tree on $n$ vertices. We can regard the edges of $T$ as transpositions of the vertex set; their product (in any order) is a cyclic permutation. All possible cyclic permutations arise (each exactly once) if and only if the tree…
We construct new families of completely regular codes by concatenation methods. By combining parity check matrices of cyclic Hamming codes, we obtain families of completely regular codes. In all cases, we compute the intersection array of…