Generalized Orthogonality Relation for Spherical Harmonics

International audienc

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

Relaxation dynamics of a flexible rod in a fluid

peer reviewe

Essential role of papillae flexibility in nectar capture by bees

peer reviewe

On quasi-orthogonal polynomials

AbstractChihara [On quasi orthogonal polynomials, Proc. Amer. Math. Soc. 8 (1957), 765–767] has shown that quasi-orthogonal polynomials satisfy a three-term recurrence relation with polynomial coefficients. In this paper it is shown that, if a sequence of polynomial coefficients is given with some particular properties, then there exists a unique sequence of monic polynomials ({Un}nϵN and U0 = 1} which satisfies a three-term recurrence relation whose polynomial coefficients are those given. The polynomials are quasi-orthogonal of order 1 with respect to a unique linear functional of moments. Some new properties of the quasi-orthogonal polynomials of order 1 are also proved

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

International audienc

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

International audienc