Related papers: The logarithmic fibbinomial formula
We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…
We give an explicit formula for the HOMFLY polynomial of a rational link (in particular, a knot) in terms of a special continued fraction for the rational number that defines the given link.
We identify quotient polynomial rings isomorphic to the recently found fundamental fusion algebras of logarithmic minimal models.
For an arbitrary homogeneous linear recurrence sequence of order d with constant coefficients, we derive recurrence relations for all subsequences with indices in arithmetic progression. The coefficients of these recurrences are given…
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…
The q-binomial coefficients are the polynomial cousins of the traditional binomial coefficients, and a number of identities for binomial coefficients can be translated into this polynomial setting. For instance, the familiar vanishing of…
A new kind of deformed calculus was introduced recently in studying of parabosonic coordinate representation. Based on this deformed calculus, a new deformation of Legendre polynomials is proposed in this paper, some properties and…
In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.
We give a formula for the coefficients of the Yablonskii-Vorob'ev polynomial. Also the reduction modulo a prime number of the polynomial is studied.
The idea of orthogonal polynomials has been generalized in two ways to achieve new types of polynomials: noncommutative orthogonal polynomials and biorthogonal polynomials. This paper brings these two different generalizations together to…
We study a natural analogue of Collatz's Conjecture for polynomials over $\mathbb{F}_2$.
We survey various classical results on invariants of polynomials, or equivalently, of binary forms, focussing on explicit calculations for invariants of polynomials of degrees 2, 3, 4.
We show that the logarithm $\log_q$ of the Frobenius morphism $x\to x^q$ is given by the formula $x\to x\log x$ (the natural logarithm). In particular, it does not depend on $q$. This is the explicit (although heuristical) formula for the…
We interpret the coefficients of the cyclotomic polynomial in terms of simplicial homology.
An investigation of classical fields with fractional derivatives is presented using the fractional Hamiltonian formulation. The fractional Hamilton's equations are obtained for two classical field examples. The formulation presented and the…
We review results of papers written on the topic of polynomial amoebas with an emphasis on computational aspects of the topic. The polynomial amoebas have a lot of applications in various domains of science. Computation of the amoeba for a…
Recently, the degenerate harmonic and the degenerate hyperharmonic numbers are introduced respectively as degenerate versions of the harmonic and the hyperharmonic numbers. The aim of this paper is to introduce the degenerate…
We give an explicit formula for the Jones polynomial of any rational link in terms of the denominators of the canonical continued fraction of the slope of the given rational link.
Connected the generalized Goncharov polynomials associated to a pair ($\partial,\mathcal{Z}$) if a delta operator $\partial$ and an interpolation grid $\mathcal{Z}$, introduced by Lorentz, Tringali and Yan in [7], with the theory of…
Following Lucas and then other Fibonacci people Kwasniewski had introduced and had started ten years ago the open investigation of the overall F-nomial coefficients which encompass among others Binomial, Gaussian and Fibonomial coefficients…