A combinatorial formula for Macdonald polynomials

In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for Macdonald polynomials for all Lie types. These formulas resemble the formulas of Haglund, Haiman and Loehr for Macdonald polynomials of type GLn. At q=0 these formulas specialize to the formula of Schwer for the Macdonald spherical function in terms of positively folded alcove walks and at q=t=0 these formulas specialize to the formula for the Weyl character in terms of the Littelmann path model (in the positively folded gallery form of Gaussent and Littelmann).     

A combinatorial formula for Macdonald polynomials

We prove a combinatorial formula for the Macdonald polynomial which had been conjectured by Haglund. Corollaries to our main theorem include the expansion of in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Schützenberger for Hall-Littlewood polynomials, a new proof of Knop and Sahi's combinatorial formula for Jack polynomials as well as a lifting of their formula to integral form Macdonald polynomials, and a new combinatorial rule for the Kostka-Macdonald coefficients in the case that is a partition with parts .

     

A Chebysheff recursion formula for Coxeter polynomials

A polynomial f(T)∈Z[T] is represented by q(T)∈Z[T] if ; f(T) is graphically represented if for χ_M(T) the characteristic polynomial of a symmetric matrix M. Many instances of Coxeter polynomialsf_A(T), for A a finite dimensional algebra, are (graphically) representable. We study the case of extended canonical algebras A, see [H. Lenzing, J.A. de la Peña, Extended canonical algebras and Fuchsian singularities, in press], show that the corresponding polynomials f_A(T) are representable and satisfy a Chebysheff type recursion formula. We get consequences for the eigenvalues of the Coxeter transformation of A showing, for instance, that at most four eigenvalues may lie outside the unit circle.     

A Rodrigues-type formula for Gegenbauer matrix polynomials

This paper centers on the derivation of a Rodrigues-type formula for the Gegenbauer matrix polynomial. A connection between Gegenbauer and Jacobi matrix polynomials is given.     

A formula for polynomials with Hermitian matrix argument

We construct and study orthogonal bases of generalized polynomials on the space of Hermitian matrices. They are obtained by the Gram-Schmidt orthogonalization process from the Schur polynomials. A Berezin-Karpelevich type formula is given for these multivariate polynomials. The normalization of the orthogonal polynomials of Hermitian matrix argument and expansions in such polynomials are investigated.     

Addition formula for q-disk polynomials

Abstract. Explicit models are constructed for irreducible *-representations of the quantised universalenveloping algebra Uq(gl(n)). The irreducible decomposition of these modules with respect to thesubalgebra Uq(gl(n-1)) is given, and the corresponding spherical and associated spherical elementsare determined in terms of little q-Jacobi polynomials. This leads to a proof of an addition theoremfor the spherical elements, the so-called q-disk polynomials.     

Combinatorial formula for Macdonald polynomials and generic Macdonald polynomials

We give a direct proof of the combinatorial formula for interpolation Macdonald polynomials by introducing certain polynomials, which we call generic Macdonald polynomials, and which depend on d additional parameters and specialize to all Macdonald polynomials of degree d. The form of these generic polynomials is that of a Bethe eigenfunction and they imitate, on a more elementary level, the R-matrix construction of quantum immanants.     

A very simple proof of the eta transformation formula

We give a very simple proof of a classical transformation formula for the Dedekind eta function. This proof is a simplified version of an approach suggested by H. Petersson.     

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. To cite this article: M. Lassalle, M. Schlosser, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials

We consider 3-parametric polynomialsP ^* (x; q, t, s) which replace theA _n-series interpolation Macdonald polynomialsP ^* (x; q, t) for theBC _n-type root system. For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation. Ass the polynomialsP ^* (x; q, t, s) becomeP ^* (x; q, t). We also prove a binomial formula for 6-parametric Koornwinder polynomials.     

A combinatorial proof of a formula for the Lucas-Narayana polynomials

In 2020, Bennett, Carrillo, Machacek and Sagan gave a polynomial generalization of the Narayana numbers and conjectured that these polynomials have positive integer coefficients for $1\le k\le n$ and for $n\ge 1$. In 2020, Sagan and Tirrell used a powerful algebraic method to prove this conjecture (in fact, they extend and prove the conjecture for more than just the type A case). In this paper we give a combinatorial proof of a formula satisfied by the Lucas-Narayana polynomials described by Bennett et al. This gives a combinatorial proof that these polynomials have positive integer coefficients. A corollary of our main result establishes a parallel theorem for the FiboNarayana numbers $N_{n,k,F}$, providing a combinatorial proof of the conjecture that these are positive integers for $n\ge 1$.     

A representation formula of sum rules for zeros of polynomials

A representation formula in terms of Lucas polynomials of the second kind in several variables (see formula (4.3)), for the sum rulesJ _s ⁽ⁱ⁾ introduced by K.M. Case [1] and studied by J.S. Dehesa et al. [2]–[3] in order to obtain informations about the zeros’ distribution of eigenfunctions of a class of ordinary polynomial differential operator, is derived. Lavoro eseguito nell’ambito del G.N.I.M. del C.N.R.     

A Christoffel–Darboux formula for multiple orthogonal polynomials

Bleher and Kuijlaars recently showed that the eigenvalue correlations from matrix ensembles with external source can be expressed by means of a kernel built out of special multiple orthogonal polynomials. We derive a Christoffel–Darboux formula for this kernel for general multiple orthogonal polynomials. In addition, we show that the formula can be written in terms of the solution of the Riemann–Hilbert problem for multiple orthogonal polynomials, which will be useful for asymptotic analysis.     

New fermionic formula for unrestricted Kostka polynomials

A new fermionic formula for the unrestricted Kostka polynomials of type is presented. This formula is different from the one given by Hatayama et al. and is valid for all crystal paths based on Kirillov-Reshetikhin modules, not just for the symmetric and antisymmetric case. The fermionic formula can be interpreted in terms of a new set of unrestricted rigged configurations. For the proof a statistics preserving bijection from this new set of unrestricted rigged configurations to the set of unrestricted crystal paths is given which generalizes a bijection of Kirillov and Reshetikhin.     

A composition formula for squares of Hermite polynomials and its generalizations

We prove a general formula which, with appropriately chosen parameters, gives a composition formula for squares of Gould–Hopper polynomials g²_n(x,h), and hence also for Hermite polynomials. Our main tool is the classical Mehler formula, but with imaginary arguments. To cite this article: P. Graczyk, A. Nowak, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A product formula for orthogonal polynomials associated with infinite distance-transitive graphs

The infinite, locally finite distance-transitive graphs form an extension of homogeneous trees and are described by two discrete parameters. The associated orthogonal polynomials may be regarded as spherical functions of certain Gelfand pairs or as characters of some polynomial hypergroups; they are certain Bernstein polynomials and admit a discrete nonnegative product formula. In this paper we use the graph-theoretic origin of these polynomials to derive the existence of positive dual continuous product and transfer formulas. The dual product formulas will be computed explicitly.     

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.