Related papers: Fibonomial determinants
We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by…
We consider $q$-binomial coefficients built from the $q$-rational and $q$-real numbers defined by Morier-Genoud and Ovsienko in terms of continued fractions. We establish versions of both the $q$-Pascal identity and the $q$-binomial theorem…
We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…
We prove a new formula for the generating function of polynomials counting absolutely stable representations of quivers over finite fields. The case of irreducible representations is studied in more detail.
In this paper, we consider the problem of representing any polynomial in terms of the degenerate Bernoulli polynomials and more generally of the higher-order degenerate Bernoulli polynomials. We derive explicit formulas with the help of…
A thorough analysis is made of the Fourier coefficients for vector-valued modular forms associated to three-dimensional irreducible representations of the modular group. In particular, the following statement is verified for all but a…
The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their…
We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…
We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the…
We prove an interesting identity for the sum of determinants, which is a generalization of the sum of a geometric progression. The proof is quite long and a number of other identities are proved along the way. Some of the more elementary…
Using continued fraction expansions of certain polygamma functions as a main tool, we find orthogonal polynomials with respect to the odd-index Bernoulli polynomials $B_{2k+1}(x)$ and the Euler polynomials $E_{2k+\nu}(x)$, for $\nu=0, 1,…
We combine Young idempotents in the group algebra of the symmetric group with the action of the symmetric group on products of Vandermonde determinants to obtain idempotents with polynomial coefficients.
The paper is mostly a survey on recent results in Diophantine approximation, with emphasis on properties of exponents measuring various notions of Diophantine <approximation.
We construct biorthogonal polynomials for a measure over the complex plane which consists in the exponential of a potential V(z,z*) and in a set of external sources at the numerator and at the denominator. We use the pseudonorm of these…
The main aim of this paper is to define and investigate a new class of the degenerate poly-Frobenius-Genocchi polynomials with the help of the polyexponential functions. In this paper, we define the degenerate poly-Frobenius-Genocchi…
Certain classical generating functions for elements of reflection groups can be expressed using fundamental invariants called exponents. We give new analogues of such generating functions that accommodate orbits of reflecting hyperplanes…
We discuss several conjectures about the real-rootedness of polynomials whose coefficients are determinants of coefficients of a real-rooted polynomial. We also consider some questions about matrices generalizing totally positive matrices,…
This is an indicatory presentation of main definitions and theorems of fibonomial calculus which is a special case of psi-extented rota's finite operator calculus.
In this paper, we consider the matrix polynomial obtained by using bi-periodic Fibonacci matrix polynomial. Then, we give some properties and binomial transforms of the new matrix polynomials.
We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct…