Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm

The discrete-time Toda equation arises as a universal equation for the relevant Hankel determinants associated with one-variable orthogonal polynomials through the mechanism of adjacency, which amounts to the inclusion of shifted weight functions in the orthogonality condition. In this paper we extend this mechanism to a new class of two-variable orthogonal polynomials where the variables are related via an elliptic curve. This leads to a `Higher order Analogue of the Discrete-time Toda' (HADT) equation for the associated Hankel determinants, together with its Lax pair, which is derived from the relevant recurrence relations for the orthogonal polynomials. In a similar way as the quotient-difference (QD) algorithm is related to the discrete-time Toda equation, a novel quotient-quotient-difference (QQD) scheme is presented for the HADT equation. We show that for both the HADT equation and the QQD scheme, there exists well-posed $s$-periodic initial value problems, for almost all \s\in\Z^2. From the Lax-pairs we furthermore derive invariants for corresponding reductions to dynamical mappings for some explicit examples.Comment: 38 page

Δ-Coherent pairs of linear functionals and Markov-Bernstein inequalities

International audienceAll the linear functionals c0 and c1 associated to the Δ-coherent pairs (c0,c1), have been given by Area et al. (2000). From these linear functionals c0 and c1 are introduced bilinear functionals aλ(p,q)=c0(pq)+λc1(ΔpΔq), ∀p,q∈P, but only for the Δ-coherent pairs (c0,c1) having a support Ω=]0,+∞[. Five kinds of Δ-coherent pairs given by Area et al. are concerned. They imply Charlier polynomials or Meixner polynomials, and they depend on a set of parameters. This paper is devoted to the study of the positivity of aλ(p,p) in order to obtain Markov–Bernstein inequalitiesc1((Δp)2)≤M2nc0(p2),∀p∈Pn.The Markov–Bernstein constant Mnis equal to 1μ1,n√ where μ1,n is the smallest zero of a polynomial of degree n satisfying a three term recurrence relation. The five kinds of three term recurrence relation are obtained. The behavior of μ1,n is studied for a part of the variation of the parameters which characterize the Δ-coherent pairs

On Asymptotics of the Sharp Constants of the Markov–Bernshtein Inequalities for the Sobolev Spaces

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Asymptotics of sharp constants of Markov-Bernstein inequalities in integral norm with Jacobi weight

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Two spaces of generalized functions based on harmonic polynomials

Two spaces of generalized functions on the unit sphere Ωq−1 ⊂ ℝq are introduced. Both types of generalized functions can be identified with suitable classes of harmonic functions. They are projective and inductive limits of Hilbert spaces. Several natural classes of continuous and continuously extendible operators are discussed: Multipliers, differentiations, harmonic contractions/expansions and harmonic shifts. The latter two classes of operators are "parametrized" by the full affine semigroup ℝn. AMS Classifications 46F05 46F10 31B05 20G0