Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions

Suppose is a sequence of polynomials orthogonal with respect to the moment functional τ=σ+ν, where σ is a classical moment functional (Jacobi, Laguerre, Hermite) and ν is a point mass distribution with finite support. In this paper, we develop a new method for constructing a differential equation having as eigenfunctions.     

Differential operators having Sobolev type Laguerre polynomials as eigenfunctions

We consider the polynomials orthogonal with respect to the Sobolev type inner product

where and is a nonnegative integer. It is the purpose of this paper to show that these polynomials are eigenfunctions of a class of linear differential operators containing one that is of finite order if is a nonnegative integer and

     

Chain sequences and symmetric generalized orthogonal polynomials

In this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K_n^(λ,M,k) associated with the probability measure dφ(λ,M,k;x), which is the Gegenbauer measure of parameter λ+1 with two additional mass points at ±k. When k=1 we obtain information on the polynomials K_n^(λ,M) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K_n^(λ,M,k) in relation to M and k are also given.     

Differential coefficients of orthogonal matrix polynomials

We find explicit formulas for raising and lowering first order differential operators for orthogonal matrix polynomials. We derive recurrence relations for the coefficients in the raising and lowering operators. Some examples are given.     

Linearization of the product of symmetric orthogonal polynomials

A new constructive approach is given to the linearization formulas of symmetric orthogonal polynomials. We use the monic three-term recurrence relation of an orthogonal polynomial system to set up a partial difference equation problem for the product of two polynomials and solve it in terms of the initial data. To this end, an auxiliary function of four integer variables is introduced, which may be seen as a discrete analogue of Riemann's function. As an application, we derive the linearization formulas for the associated Hermite polynomials and for their continuousq-analogues. The linearization coefficients are represented here in terms of₃ F ₂ and₃Φ₂ (basic) hypergeometric functions, respectively. We also give some partial results in the case of the associated continuousq-ultraspherical polynomials.     

Some examples of matrix-valued orthogonal functions having a differential and an integral operator as eigenfunctions

The aim of this paper is to show some examples of matrix-valued orthogonal functions on the real line which are simultaneously eigenfunctions of a second-order differential operator of Schrödinger type and an integral operator of Fourier type. As a consequence we derive integral equations of these functions as well as other useful structural formulas. Some of these functions are plotted to show the relationship with the Hermite or wave functions.     

Intertwining operators and polynomials associated with the symmetric group

There is an algebra of commutative differential-difference operators which is very useful in studying analytic structures invariant under permutation of coordinates. This algebra is generated by the Dunkl operators , (i=1, ...,N, where (ij) denotes the transposition of the variablesx _i x _j andk is a fixed parameter). We introduce a family of functions {p }, indexed bym-tuples of non-negative integers = (₁, ..., _m ) formN, which allow a workable treatment of important constructions such as the intertwining operatorV. This is a linear map on polynomials, preserving the degree of homogeneity, for which ,i = 1, ...,N, normalized byV1=1 (seeDunkl, Canadian J. Math.43 (1991), 1213–1227). We show thatT _i p =0 fori>m, and

The symmetric D ω -semi-classical orthogonal polynomials of class one

We give the system of Laguerre–Freud equations associated with the D _ω -semi-classical functionals of class one, where D _ω is the divided difference operator. This system is solved in the symmetric case. There are essentially two canonical cases. The corresponding integral representations are given.     

Differential equations of infinite order for orthogonal polynomials

Summary Differential equations of infinite order of the form , where M_k is a polynomial of degree ≦k, have polynomial solutions for suitable values of λ. Necessary and sufficient conditions are given in order that these solutions form a set of orthogonal polynomials. The cases where { degree M_k }=1 and 2 are studied in detail. Supported by N.S.F. Grant GP-5311.     

Concentration of symmetric eigenfunctions

In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures corresponding to eigenfunctions. These measures describe simultaneously the concentration and oscillation effects developed by a sequence of eigenfunctions. We present some results showing how to obtain invariant semiclassical measures from eigenfunctions with prescribed symmetries. As an application of these results, we give a simple proof of the fact that in a manifold of constant positive sectional curvature, every measure which is invariant by the geodesic flow is an invariant semiclassical measure.     

Differential operators associated with zonal polynomials. I

Summary Associated with each zonal polynomial,C _k(S), of a symmetric matrixS, we define a differential operator ∂_k, having the basic property that ∂_kC_λδ_kλ, δ being Kronecker's delta, whenever κ and λ are partitions of the non-negative integerk. Using these operators, we solve the problems of determining the coefficients in the expansion of (i) the product of two zonal polynomials as a series of zonal polynomials, and (ii) the zonal polynomial of the direct sum,S⊕T, of two symmetric matricesS andT, in terms of the zonal polynomials ofS andT. We also consider the problem of expanding an arbitrary homogeneous symmetric polynomial,P(S) in a series of zonal polynomials. Further, these operators are used to derive identities expressing the doubly generalised binomial coefficients ( _P ^λ ),P(S) being a monomial in the power sums of the latent roots ofS, in terms of the coefficients of the zonal polynomials, and from these, various results are obtained.     

Differential operators associated with zonal polynomials. II

Summary LetC _κ(S) be the zonal polynomial of the symmetricm×m matrixS=(s_ij), corresponding to the partition κ of the non-negative integerk. If ∂/∂S is them×m matrix of differential operators with (i, j)th entry ((1+δ_ij)∂/∂s_ij)/2, δ being Kronecker's delta, we show that C_k(∂/∂S)C_λ(S)=k!δ_λkC_k(I), where λ is a partition ofk. This is used to obtain new orthogonality relations for the zonal polynomials, and to derive expressions for the coefficients in the zonal polynomial expansion of homogenous symmetric polynomials.     

Laplace and Segal-Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials

Let be a complex bounded symmetric domain of tube type in a complex Jordan algebra V and let be its real form in a formally real Euclidean Jordan algebra J⊂V; is a bounded realization of the symmetric cone in J. We consider representations of H that are gotten by the generalized Segal-Bargmann transform from a unitary G-space of holomorphic functions on to an L²-space on . We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extension to of the spherical functions on and find their expansion in terms of the L-spherical polynomials on , which are Jack symmetric polynomials. We prove that the coefficients are orthogonal polynomials in an L²-space, the measure being the Harish-Chandra Plancherel measure multiplied by the symbol of the Berezin transform. We prove the difference equation and recurrence relation for those polynomials by considering the action of the Lie algebra and the Cayley transform on the polynomials on . Finally, we use the Laplace transform to study generalized Laguerre functions on symmetric cones.     

Some inequalities for symmetric functions and an application to orthogonal polynomials

We present some sharp inequalities for symmetric functions and give an application to orthogonal polynomials.     

ON THE MODIFICATIONS OF CLASSICAL ORTHOGONAL POLYNOMIALS: THE SYMMETRIC CASE

We consider the modifications of the monic Hermite and Gegenbauer polynomials via the addition of one point mass at the origin. Some properties of the resulting polynomials are studied ; three-term recurrence relation, differential equation, ratio asymptotics, hypergeometric representation as well as, for large n, the behaviour of their zeros.     

A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions

In this research, by applying the extended Sturm-Liouville theorem for symmetric functions, a basic class of symmetric orthogonal polynomials (BCSOP) with four free parameters is introduced and all its standard properties, such as a generic second order differential equation along with its explicit polynomial solution, a generic orthogonality relation, a generic three term recurrence relation and so on, are presented. Then, it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class. They are respectively the generalized ultraspherical polynomials, generalized Hermite polynomials and two other sequences of symmetric polynomials, which are finitely orthogonal on (−∞,∞) and can be expressed in terms of the mentioned class directly. In this way, two half-trigonometric sequences of orthogonal polynomials, as special sub-cases of BCSOP, are also introduced.