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In this paper, a transformation formula under modular substitutions is derived for a large class of generalized Eisenstein series. Appearing in the transformation formulae are generalizations of Dedekind sums involving the periodic…
In this paper we represent a new form of condition for the consistency of the matrix equation AXB=C. If the matrix equation AXB=C is consistent, we determine a form of general solution which contains both reproductive and non-reproductive…
The Fibonacci sequence is a sequence of numbers that has been studied for hundreds of years. In this paper, we introduce the new sequence S_{k,n} with initial conditions S_{k,0} = 2b and S_{k,1} = bk + a, which is generated by the…
We show that many existing divisibility sequences can be seen as sequences of determinants of matrix divisibility sequences, which arise naturally as Jacobian matrices associated to groups of maps on affine spaces.
Inspired from non-equilibrium statistical physics models, a general framework enabling the definition and synthesis of stationary time series with a priori prescribed and controlled joint distributions is constructed. Its central feature…
In this paper, we give some determinantal and permanental representations of Generalized Fibonacci Polynomials by using various Hessenberg matrices. These results are general form of determinantal and permanental representations of k…
A Redheffer--type matrix with Fibonacci entries is defined, and the determinant and spectral properties of this matrix are studied. Also, more general Redheffer--type matrices are considered and intriguing number-theoretic examples are…
In this paper, using some arithmetic properties of Jacobi sums, we investigate some products involving Jacobi sums and reveal the connections between these products and certain cyclotomic matrices. In particular, as an application of our…
The study of bipartite maps (or Grothendieck's dessins d'enfants) is closely connected with geometry, mathematical physics and free probability. Here we study these objects from their permutation factorization formulation using a novel…
We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These…
A recursive method is given for finding generating functions which enumerate rooted hypermaps by number of vertices, edges and faces for any given number of darts. It makes use of matrix-integral expressions arising from the study of…
The recurrence matrix relations, differentiation formulas, and analytical and fractional integral properties of incomplete gamma matrix functions $\gamma(Q, x)$ and $\Gamma(Q, x)$ are all covered in this article. The generalized incomplete…
The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions.…
It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter - Saxton equation. New matrix and scalar Lax representation is presented for this generalization. New class of the…
Properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobi's quartic theta function identity and…
We give recurrence relations for any family of generalized Appell polynomials unifying so some known recurrences of many classical sequences of polynomials. Our main tool to get our goal is the Riordan group. We use the product of Riordan…
In this paper we study ternary algebras of third-order hypermatrices. By hypermatrix we mean a complex-valued variable with three indices, which is also called a three-dimensional matrix or spatial matrix. We assume that a hypermatrix is…
The work considers the set $\Lambda_n^k$ of all $n\times n$ binary matrices having the same number of $k$ units in each row and each column. The article specifically focuses on the matrices whose rows and columns are sorted…
Various specifiable combinatorial structures, with d extensive parameters, can be exactly sampled both by the recursive method, with linear arithmetic complexity if a heavy preprocessing is performed, or by the Boltzmann method, with…
We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic…