Some combinatorial conjectures for Jack polynomials

We present several combinatorial conjectures related to the expansion of Jack polynomials in terms of power sums.     

Symmetric Jack polynomials from non-symmetric theory

We show how a number of fundamental properties of the symmetric and anti-symmetric Jack polynomials can be derived from knowledge of the corresponding properties of the nonsymmetric Jack polynomials. These properties include orthogonality relations, normalization formulas, a specialization formula and the evaluation of a proportionality constant relating the anti-symmetric Jack polynomials to a product of differences multiplied by the symmetric Jack polynomials with a shifted parameter.This work was supported by the Australian Research Council.     

Classical orthogonal polynomials: A functional approach

We characterize the so-called classical orthogonal polynomials (Hermite, Laguerre, Jacobi, and Bessel) using the distributional differential equation D(u)=u. This result is naturally not new. However, other characterizations of classical orthogonal polynomials can be obtained more easily from this approach. Moreover, three new properties are obtained.     

Distributional equation for Laguerre-Hahn functionals on the unit circle

Let u be a Hermitian linear functional defined in the linear space of Laurent polynomials and F its corresponding Carathéodory function. We establish the equivalence between a Riccati differential equation with polynomial coefficients for F, zAF^′=BF²+CF+D, and a distributional equation for u, , where L is the Lebesgue functional, and the polynomials are defined in terms of the polynomials A,B,C,D.     

Asymptotic behaviour of Laguerre-Sobolev-type orthogonal polynomials. A nondiagonal case

In this paper we study the asymptotic behaviour of polynomials orthogonal with respect to a Sobolev-type inner product

     

Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions

A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in , the Jack parameter shifted by . More strongly, we make the Matchings-Jack Conjecture, that the coefficients are counting series in for matchings with respect to a parameter of nonbipartiteness. Evidence is presented for these conjectures and they are proved for two infinite families.

The coefficients of a second series, essentially the logarithm of the first, specialize at values and of the Jack parameter to the numbers of hypermaps in orientable and locally orientable surfaces, respectively. We conjecture that these coefficients are also nonnegative integer polynomials in , and we make the Hypermap-Jack Conjecture, that the coefficients are counting series in for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability.

     

Asymptotic properties of generalized Laguerre orthogonal polynomials

In the present paper we deal with the polynomials L_n^(α,M,N) (x) orthogonal with respect to the Sobolev inner product

     

A note on the zeros of Freud–Sobolev orthogonal polynomials

We prove that the zeros of a certain family of Sobolev orthogonal polynomials involving the Freud weight function e^-x4${\mathrm{e}}^{-x^4}$ on R$\mathbb{R}$ are real, simple, and interlace with the zeros of the Freud polynomials, i.e., those polynomials orthogonal with respect to the weight function e^-x4${\mathrm{e}}^{-x^4}$. Some numerical examples are shown.     

A set of orthogonal polynomials induced by a given orthogonal polynomial

Summary Given an integern 1, and the orthogonal polynomials _n (·; d) of degreen relative to some positive measured, the polynomial system induced by _n is the system of orthogonal polynomials corresponding to the modified measure . Our interest here is in the problem of determining the coefficients in the three-term recurrence relation for the polynomials from the recursion coefficients of the orthogonal polynomials belonging to the measured. A stable computational algorithm is proposed, which uses a sequence ofQR steps with shifts. For all four Chebyshev measuresd, the desired coefficients can be obtained analytically in closed form. For Chebyshev measures of the first two kinds this was shown by Al-Salam, Allaway and Askey, who used sieved orthogonal polynomials, and by Van Assche and Magnus via polynomial transformations. Here, analogous results are obtained by elementary methods for Chebyshev measures of the third and fourth kinds. (The same methods are also applicable to the other two Chebyshev measures.) Interlacing properties involving the zeros of _n and those of are studied for Gegenbauer measures, as well as the orthogonality—or lack thereof—of the polynomial sequence .Work supported in part by the National Science Foundation under grant DMS-9023403.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Meixner polynomials in several variables satisfying bispectral difference equations

We construct a set _Md${\mathfrak{M}}_d$ whose points parametrize families of Meixner polynomials in d   variables. There is a natural bispectral involution b$\mathfrak{b}$ on _Md${\mathfrak{M}}_d$ which corresponds to a symmetry between the variables and the degree indices of the polynomials. We define two sets of d   commuting partial difference operators diagonalized by the polynomials. One of the sets consists of difference operators acting on the variables of the polynomials and the other one on their degree indices, thus proving their bispectrality. The two sets of partial difference operators are naturally connected via the involution b$\mathfrak{b}$.     

When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

Given {P_n}_n≥0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e., 

     

Zeros of linear combinations of Laguerre polynomials from different sequences

We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely and . Proofs and numerical counterexamples are given in situations where the zeros of R_n, and S_n, respectively, interlace (or do not in general) with the zeros of , , k=n or n−1. The results we prove hold for continuous, as well as integral, shifts of the parameter α.     

The structure relation for Askey–Wilson polynomials

An explicit structure relation for Askey–Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey–Wilson inner product and which sends polynomials of degree n   to polynomials of degree n+1$n+1$. By specialization of parameters and by taking limits, similar structure relations, as well as lowering and raising relations, can be obtained for other families in the q-Askey scheme and the Askey scheme. This is explicitly discussed for Jacobi polynomials, continuous q-Jacobi polynomials, continuous q-ultraspherical polynomials, and for big q-Jacobi polynomials. An already known structure relation for this last family can be obtained from the new structure relation by using the three-term recurrence relation and the second order q-difference formula. The results are also put in the framework of a more general theory. Their relationship with earlier work by Zhedanov and Bangerezako is discussed. There is also a connection with the string equation in discrete matrix models and with the Sklyanin algebra.     

Pieri-type formulas for the non-symmetric Jack polynomials

In the theory of symmetric Jack polynomials the coefficients in the expansion of the $p$th elementary symmetric function $e_p(z)$ times a Jack polynomial expressed as a series in Jack polynomials are known explicitly. Here analogues of this result for the non-symmetric Jack polynomials $E_\eta(z)$ are explored. Necessary conditions for non-zero coefficients in the expansion of $e_p(z) E_\eta(z)$ as a series in non-symmetric Jack polynomials are given. A known expansion formula for $z_i E_\eta(z)$ is rederived by an induction procedure, and this expansion is used to deduce the corresponding result for the expansion of $\prod_{j=1, \, j\ne i}^N z_j \, E_\eta(z)$, and consequently the expansion of $e_{N-1}(z) E_\eta(z)$. In the general $p$ case the coefficients for special terms in the expansion are presented.     

Orthogonal polynomials on the disc

We consider the space P_n of orthogonal polynomials of degree n on the unit disc for a general radially symmetric weight function. We show that there exists a single orthogonal polynomial whose rotations through the angles , j=0,1,…,n forms an orthonormal basis for P_n, and compute all such polynomials explicitly. This generalises the orthonormal basis of Logan and Shepp for the Legendre polynomials on the disc.Furthermore, such a polynomial reflects the rotational symmetry of the weight in a deeper way: its rotations under other subgroups of the group of rotations forms a tight frame for P_n, with a continuous version also holding. Along the way, we show that other frame decompositions with natural symmetries exist, and consider a number of structural properties of P_n including the form of the monomial orthogonal polynomials, and whether or not P_n contains ridge functions.     

A characterization of the classical orthogonal discrete and q-polynomials

In this paper we present a new characterization for the classical discrete and q$q$-classical (discrete) polynomials (in the Hahn's sense).     

Generalizations of the image conjecture and the Mathieu conjecture

We first propose a generalization of the image conjecture Zhao (submitted for publication) [31] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture Mathieu (1997) [21] from integrals of G-finite functions over reductive Lie groups G to integrals of polynomials over open subsets of Rⁿ with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of the Duistermaat and van der Kallen theorem Duistermaat and van der Kallen (1998) [14] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly introduced notion.