Related papers: On coefficients of Yablonskii-Vorob'ev polynomial
We prove an explicit Chinese Remainder Theorem for one variable polynomials with complex coefficients, and derive some consequences.
Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is…
We exhibit a numerical method to solve fractional variational problems, applying a decomposition formula based on Jacobi polynomials. Formulas for the fractional derivative and fractional integral of the Jacobi polynomials are proven. By…
Jacobi-Trudy formula for a generalisation of Schur polynomials related to any sequence of orthogonal polynomials in one variable is given. As a corollary we have Giambelli formula for generalised Schur polynomials.
In this paper, we gave some properties of binomial coefficient.
It has been known that, the coefficients of the series expansion of the Jones polynomial evaluated at $e^x$ are rational valued Vassiliev invariants . In this article, we calculate minimal multiplying factor, {\lambda}, needed for these…
In this paper, we introduce the notion of Jacobi polynomials with multiple reference vectors of a code, and give the MacWilliams type identity for it. Moreover, we derive a formula to obtain the Jacobi polynomials using the Aronhold…
There are several reformulations of the Vi\`ete's formula for pi that have been reported in the modern literature. In this paper we show another analog to the Vi\`ete's formula for pi by Chebyshev polynomials of the first kind.
In this paper, we study a class of orthogonal polynomials defined by a three-term recurrence relation with periodic coefficients. We derive explicit formulas for the generating function, the associated continued fraction, the orthogonality…
The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Jacobi-Sobolev bilinear form with mass point at $-1$ and/or $+1$. In particular, we construct the orthogonal polynomials using…
We obtain an explicit combinatorial formula for certain parabolic Kostka-Shoji polynomials associated with the cyclic quiver, generalizing results of Shoji and of Liu and Shoji.
The kernel polynomial method based on Jacobi polynomials $P_n^{\alpha,\beta}(x)$ is proposed. The optimal-resolution positivity-preserving kernels and the corresponding damping factors are obtained. The results provide a generalization of…
We state a general formula for the number of binomial coefficients $n$ choose $k$ that are divided by a fixed power of a prime $p$, i.e., the number of binomial coefficients divided by $p^j$ and not divided by $p^{j+1}$.
In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…
We use a combinatorial interpretation of the coefficients of zonal Kerov polynomials as a number of unoriented maps to derive an explicit formula for the coefficients in genus one.
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 compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture using a recursive formula for the Catalan numbers.
The main aim of this paper is to find the estimation for Vilenkin-Fourier coefficients.
A class P_{n,m,p}(x) of polynomials is defined. The combinatorial meaning of its coefficients is given. Chebyshev polynomials are the special cases of P_{n,m,p}(x). It is first shown that P_{n,m,p}(x) may be expressed in terms of…
We shall give an explicit version of Bombieri-Vinogradov Theorem for moduli not divisible by an exceptional modulus.