An analytic formula for Macdonald polynomials

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.Comment: 8 pages; research announcement submitted to Comptes Rendus Math. Acad. Sci. Paris for publicatio

Jack polynomials and some identities for partitions

We prove an identity about partitions involving new combinatorial coefficients. The proof given is using a generating function. As an application we obtain the explicit expression of two shifted symmetric functions, related with Jack polynomials. These quantities are the moments of the "alpha-content" random variable with respect to some transition probability distributions.Comment: 22 pages, LaTeX, to appear in Trans. Amer. Math. So

Inversion of the Pieri formula for Macdonald polynomials

We give the explicit analytic development of Macdonald polynomials in terms of "modified complete" and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments for monomial, Jack and Hall-Littlewood symmetric functions.Comment: 34 page

Two positivity conjectures for Kerov polynomials

Kerov polynomials express the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as polynomials in the free cumulants of the associated Young diagram. We present two positivity conjectures for their coefficients. The latter are stronger than the positivity conjecture of Kerov-Biane, recently proved by Feray.Comment: 15 pages, LaTeX, final version, to appear in Adv. Appl. Mat

Class expansion of some symmetric functions in Jucys-Murphy elements

We present a method to compute the class expansion of a symmetric function in the Jucys-Murphy elements of the symmetric group. We apply this method to one-row Hall-Littlewood symmetric functions, which interpolate between power sums and complete symmetric functions.Comment: 53 pages, LaTeX, to appear in Journal of Algebr

A new family of positive integers

Let n,p,k be three positive integers. We prove that the numbers binomial (n,k) 3F2 (1-k, -p, p-n ; 1, 1-n ; 1) are positive integers which generalize the classical binomial coefficients. We give two generating functions for these integers, and a straightforward application.Comment: Enlarged version, LaTeX, 7 page

A Conjecture about Raising Operators for Macdonald Polynomials

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 and Schlosser's formula for partitions with length three.Comment: 13 page