Abstract. Integrability, algebraic structures and orthogonal basis of the Calogero model are studied by the quantum Lax and Dunkl operator formulations. The commutator algebra among operators including conserved operators and creation-annihilation operators has the structure of the W-algebra. Through an algebraic construction of the simultaneous eigenfunctions of all the commuting conserved ope...

In this note we provide an algorithm for computing the fractional integrals of orthogonal polynomials, which is more stable than one based on expansion polynomials w.r.t. canonical basis. This aimed at solving corresponding differential equations. A few numerical illustrations are reported.

in this paper we establish several polynomials similar to bernstein's polynomials and several refinements of  hermite-hadamard inequality for convex functions.

We consider the space Pn of orthogonal polynomials of degree n on the unit disc for a general radially symmetricweight function.We show that there exists a single orthogonal polynomialwhose rotations through the angles j n+1 , j = 0, 1, . . . , n forms an orthonormal basis for Pn, and compute all such polynomials explicitly. This generalises the orthonormal basis of Logan and Shepp for the Lege...

These notes will attempt to collect a lot of information about polynomials and Stirling numbers and the calculus of finite differences. Let Pn denote the set of all polynomials of degree ≤ n. It is easy to see that this is a vector space over the reals (or the complex numbers if you like) of dimension n+ 1. The most familiar basis for Pn is: {1, x, x, . . . , x}. Any p ∈ Pn is almost instantane...

Young’s lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing lattice, we introduce new weak compositions call key poset. Saturated in this poset objects generate polynomials, nonsymmetric polynomial generalization basis. skew functions, ...

In this paper we consider a particularly important case of 3D monogenic polynomials that are isomorphic to the integer powers of one complex variable (called pseudo-complex powers or pseudo-complex polynomials, PCP). The construction of bases for spaces of monogenic polynomials in the framework of Clifford Analysis has been discussed by several authors and from different points of view. Here ou...

Standard noncommutative Gröbner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields. This paper describes Gröbner basis procedures for one-sided ideals in finitely presented noncommutative algebras over fields. The polynomials defining a K-algebra A as a quotient of a free K-algebra are combined with the polynomials defining a one-sided ideal I of ...

A monomial orthogonal polynomial of several variables is of the form x−Qα(x) for a multiindex α ∈ N 0 and it has the least L2 norm among all polynomials of the form xα − P (x), where P and Qα are polynomials of degree less than the total degree of xα. We study monomial orthogonal polynomials with respect to the weight function ∏d+1 i=1 |xi| 2κi on the unit sphere Sd as well as for the related w...