We consider the glN -invariant Calogero-Sutherland Models with N = 1, 2, 3, . . . in a unified framework, which is the framework of Symmetric Polynomials. By the framework we mean an isomorphism between the space of states of the glN -invariant Calogero-Sutherland Model and the space of Symmetric Laurent Polynomials. In this framework it becomes apparent that all the glN -invariant Calogero-Sut...

In this paper, we consider a novel family of the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials. This represents fascinating fusion between two distinct categories special functions: Bernoulli polynomials and Gegenbauer We focus our attention on some algebraic differential properties class polynomials, including its explicit expressions, derivative formulas, matrix representations, ...

The one-parameter family of polynomials Jn(x, y) = ∑n j=0 ( y+j j ) x is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each n ≥ 6, the polynomial Jn(x, y0) is irreducible over Q for all but finitely many y0 ∈ Q. If n is odd, then with the exception of a finite set of y0, the Galois group of Jn(x, y0) is Sn; if n is even, then the exceptional set is thin.

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several generalizations of formula for generalized functions polynomials. As a consequence, give new addition an integral representation these Finally, introduce family Lebesgue spaces show that some special belong to them.

In "An almost Cubic Lower Bound for Depth Three Arithmetic Circuits", [KST16] present an infinite family of polynomials in VNP, {Pn}n∈Z+ on n variables with degree n such that every ∑∏∑ circuit computing Pn is of size Ω̃(n3). A similar result was proven in [BLS16] for polynomials in VP with lower bound Ω ( n3 2 p logn ) . We present a modified polynomial and perform a tighter analysis to obtain ...

We consider multivariate polynomials with exponents that are themselves integer-valued multivariate polynomials, and we present algorithms to compute their GCD and factorization. The algorithms fall into two families: algebraic extension methods and projection methods. The first family of algorithms uses the algebraic independence of x, x, x 2 , x, etc, to solve related problems with more indet...

Despite the tremendous development in CNC programming facilities, linear and circular cuts parallel to the coordinate planes continue to be the standard motions of modern CNC machines. However, the increasing industrial demand for parts with intricate shapes cannot be satisfied with only these standard motions. The proportion of parts which are not covered by the standard CNC motions is certain...

We study a family of polynomials introduced by Daigle and Freudenburg, which contains the famous Vénéreau defines

Knot polynomials colored with symmetric representations of $SL_q(N)$ satisfy difference equations as functions representation parameter, which look like quantization classical ${\cal A}$-polynomials. However, they are quite difficult to derive and investigate. Much simpler should be the for coefficients differential expansion nicknamed quantum C}$-polynomials. It turns out that, each knot, one ...

We investigate the zeros of a family of hypergeometric polynomials 2F1(−n,−x; a; t), n ∈ N that are known as the Meixner polynomials for certain values of the parameters a and t. When a = −N, N ∈ N and t = p , the polynomials Kn(x; p,N) = (−N)n2F1(−n,−x;−N; p ), n = 0, 1, . . .N, 0 < p < 1 are referred to as Krawtchouk polynomials. We prove results for the zero location of the orthogonal polyno...