Related papers: A Rodrigues formula for the Jack polynomials and t…
Formulas of Rodrigues-type for the Macdonald polynomials are presented. They involve creation operators, certain properties of which are proved and other conjectured. The limiting case of the Jack polynomials is discussed.
We present formulas of Rodrigues type giving the Macdonald polynomials for arbitrary partitions through the repeated application of creation operators on the constant 1. Three expressions for the creation operators are derived one from the…
Classical orthogonal polynomials in one variable can be characterized as the only orthogonal polynomials satisfying a Rodrigues formula. In this paper, using the second kind Kronecker power of a matrix, a Rodrigues formula is introduced for…
Applying a method developed by Takamura and Takano for the nonsymmetric Jack polynomial, we present the Rodrigues formula for the nonsymmetric multivariable Hermite polynomial.
It is known that Rodrigues formulas provide a very powerful tool to compute orthogonal polynomials with respect to classical weights. We provide an example of bivariate multiple polynomials on the simplex defined via a Rodrigues formula.…
Jack polynomials generalize several classical families of symmetric polynomials, including Schur polynomials, and are further generalized by Macdonald polynomials. In 1989, Richard Stanley conjectured that if the Littlewood-Richardson…
We present a positivity conjecture for the coefficients of the development of Jack polynomials in terms of power sums. This extends Stanley's ex-conjecture about normalized characters of the symmetric group. We prove this conjecture for…
We give an explicit formula for the power-sum expansion of Jack polynomials. We deduce it from a more general formula, which we provide here, that interprets Jack characters in terms of bipartite maps. We prove Lassalle's conjecture from…
Heckman and Opdam introduced a non-symmetric analogue of Jack polynomials using Cherednik operators. In this paper, we derive a simple recursion formula for these polynomials and formulas relating the symmetric Jack polynomials with the…
In this note we prove an explicit binomial formula for Jack polynomials and discuss some applications of it.
A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a…
We prove a previously conjectured closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of…
A multivariable hypergeometric-type formula for raising operators of the Macdonald polynomials is conjectured. It is proved that this agrees with Jing and Jozefiak's expression for the two-row Macdonald polynomials, and also with Lassalle…
The Hi-Jack symmetric polynomials, which are associated with the simultaneous eigenstates for the first and second conserved operators of the quantum Calogero model, are studied. Using the algebraic properties of the Dunkl operators for the…
This paper surveys eight classes of polynomials associated with $A$-type and $BC$-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and interpolation (or shifted) Jack and Macdonald polynomials and their $BC$-type…
The properties of matrix valued polynomials generated by the scalar-type Rodrigues' formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence…
Necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials on lattices are stated. Moreover, the functional Rodrigues formula and a closed…
We give the explicit analytic development of any Jack or Macdonald polynomial in terms of elementary (resp. modified complete) symmetric functions. These two developments are obtained by inverting the Pieri formula.
We present a simple formula for the generating function for the polynomials in the $d$--dimensional semiclassical wave packets. We then use this formula to prove the associated Rodrigues formula.
We prove a binomial formula for Macdonald polynomials and consider applications of it.