A different approach to Gauss Fibonacci polynomials

In this paper with the help of higher order Fibonacci polynomials, we introduce higher order Gauss Fibonacci polynomials that generalize the Gauss Fibonacci polynomials studied by Özkan and Taştan. We give a recurrence relation, Binet-like formula, generating and exponential generating functions, summation formula for the higher order Gauss Fibonacci polynomials. Moreover, we give two special matrices that we call $Q^{(s)}(x)$ and $P^{(s)}(x),$ respectively. From these matrices, we obtain a matrix representation and derive the Cassini's identity of higher order Gauss Fibonacci polynomials

Moments of qq-Jacobi polynomials and qq-zeta values

We explore some connections between moments of rescaled little $q$-Jacobi polynomials, $q$-analogues of values at negative integers for some Dirichlet series, and the $q$-Eulerian polynomials of wreath products of symmetric groups

Two families of strongly walk regular graphs from three-weight codes over Z4\mathbb{Z_4}

A necessary condition for a $\mathbb{Z_4}$-code to be a three-weight code for the Lee weight is given. Two special constructions of three-weight codes over $\mathbb{Z_4}$ are derived. The coset graphs of their duals are shown to be strongly 3-walk-regular, a generalization of strongly regular graphs

On the permanent of an even-dimensional non-negative polystochastic tensor of order n

In this paper, we present an algorithm that allows us to compute the permanent of a tensor by using Laplace expansion. We prove that the permanent of a $4$-dimensional polystochastic $(0,1)$-tensor of order $n$ constructed using a special $n\times (n-1)$ row-Latin rectangle $R$ with no transversals is positive. Also, we show that the permanent of an even-dimensional polystochastic $(0,1)$-tensor of order $n$ constructed using the row-Latin rectangle $R$ is positive. The result obtained here proves that each odd-dimensional Latin hypercube of order $4$ has a transversal (Wanless' conjecture for odd-dimensional Latin hypercubes of order $4$). We prove that the number of perfect matchings of the bipartite hypergraph associated to an even-dimensional polystochastic $(0,1)$-tensor of order $4$ is positive. Furthermore, we extend some results concerning polystochastic $(0,1)$-tensors to nonnegative polystochastic tensors. Moreover, we prove that the permanent of a $4$-dimensional nonnegative polystochastic tensor of order $n$ constructed using the row-Latin rectangle $R$ is positive. More generally, we show that the permanent of an even-dimensional nonnegative polystochastic tensor of order $n$ constructed using the row-Latin rectangle $R$ is positive. The result obtained here proves that the permanent of an even-dimensional nonnegative polystochastic tensor of order $4$ is positive

Hankel determinants of certain sequences of Bernoulli polynomials: A direct proof of an inverse matrix entry from statistics

We calculate the Hankel determinants of sequences of Bernoulli polynomials. This corresponding Hankel matrix comes from statistically estimating the variance in nonparametric regression. Besides its entries’ natural and deep connection with Bernoulli polynomials, a special case of the matrix can be constructed from a corresponding Vandermonde matrix. As a result, instead of asymptotic analysis, we give a direct proof of calculating an entry of its inverse. Further extensions also include an identity of Stirling numbers of the both kinds

Beta distributions whose moment sequences are related to integer sequences listed in the OEIS

We recall some basic properties of the Beta distribution and some of its modifications. We identified around $20$ of the moment sequences of Beta distributions as important integer sequences in the OEIS base of integer sequences. Among those identified are Catalan, Riordan, Motzkin, or `super ballot numbers'. By applying a method of expansion of the ratio of densities of involved distributions we are able to obtain some known and many unknown relationships between e.g. Catalan numbers and other moment sequences of the Beta distributions

The Lifting Properties for A-Homotopy Theory

In classical homotopy theory, two spaces are considered homotopy equivalent if one space can be continuously deformed into the other. This theory, however, does not respect the discrete nature of graphs. For this reason, a discrete homotopy theory that recognizes the difference between the vertices and edges of a graph was invented, called A-homotopy theory \cite{Atkin1, Atkin2, BabsonHomotopy, BarceloFoundations, BarceloPerspectives}. In classical homotopy theory, covering spaces and lifting properties are often used to compute the fundamental group of the circle. In this paper, we develop the lifting properties for A-homotopy theory. Using a covering graph and these lifting properties, we compute the fundamental group of the cycle $C_{5}$, giving an alternate approach to [4]

A linear time approach to three-dimensional reconstruction by discrete tomography

The goal of discrete tomography is to reconstruct an unknown function $f$ via a given set of line sums. In addition to requiring accurate reconstructions, it is favourable to be able to perform the task in a timely manner. This is complicated by the presence of ghosts, which allow many solutions to exist in general. In this paper we consider the case of a function $f : A \to \mathbb{R}$ where $A$ is a finite grid in $\mathbb{Z}^3$. Previous work has shown that in the two-dimensional case it is possible to determine all solutions in parameterized form in linear time (with respect to the number of directions and the grid size) regardless of whether the solution is unique. In this work, we show that a similar linear method exists in three dimensions under the condition of nonproportionality. We show that the condition of nonproportionality is fulfilled in the case of three-dimensional boundary ghosts

Unified framework for tableau models of Grothendieck Polynomials

We give combinatorial proofs of two types of duality for Grothendieck polynomials by constructing a unified combinatorial\\ framework incorporating set-valued tableaux, multiset-valued tableaux, reverse plane partitions and valued-set tableaux. Importantly, our proofs extend to proofs of these dualities for the refined Grothendieck polynomials. The second of these dualities was formerly unknown for the refined case

A new trigonometric identity with applications

In this paper we obtain a new curious identity involving trigonometric functions. Namely, for any positive odd integer $n$, we prove that $$\sum_{k=1}^n(-1)^k(\cot kx)\sin k(n-k)x=\frac{1-n}2,$$ which is equivalent to the identity $$\sum_{k=1}^n(-1)^kU_{n-k}(\cos kx)=-\frac{n+1}{2},$$ where $U_m(z)$ stands for the $m$th Chebyshev polynomial of the second kind. As a consequence, for any positive odd integer $n$ and positive integer $m$, we obtain the identity $$\sum_{k=1}^n(-1)^kk^{2m}B_{2m+1}\left(\frac{n-k}2\right)=0,$$ where $B_j(x)$ denotes the Bernoulli polynomial of degree $j$