253 research outputs found

    Mehler-Heine asymptotics for multiple orthogonal polynomials

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    Mehler-Heine asymptotics describe the behavior of orthogonal polynomials near the edges of the interval where the orthogonality measure is supported. For Jacobi polynomials and Laguerre polynomials this asymptotic behavior near the hard edge involves Bessel functions JαJ_\alpha. We show that the asymptotic behavior near the endpoint of the interval of (one of) the measures for multiple orthogonal polynomials involves a generalization of the Bessel function. The multiple orthogonal polynomials considered are Jacobi-Angelesco polynomials, Jacobi-Pi\~neiro polynomials, multiple Laguerre polynomials, multiple orthogonal polynomials associated with modified Bessel functions (of the first and second kind), and multiple orthogonal polynomials associated with Meijer GG-functions.Comment: 15 pages. Typos corrected, references updated, section "concluding remarks" adde

    Majorization results for zeros of orthogonal polynomials

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    We show that the zeros of consecutive orthogonal polynomials pnp_n and pn−1p_{n-1} are linearly connected by a doubly stochastic matrix for which the entries are explicitly computed in terms of Christoffel numbers. We give similar results for the zeros of pnp_n and the associated polynomial pn−1(1)p_{n-1}^{(1)} and for the zeros of the polynomial obtained by deleting the kkth row and column (1≤k≤n)(1 \leq k \leq n) in the corresponding Jacobi matrix.Comment: 15 page

    Compact Jacobi matrices: from Stieltjes to Krein and M(a,b)

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    In a note at the end of his paper {\it Recherches sur les fractions continues}, Stieltjes gave a necessary and sufficient condition when a continued fraction is represented by a meromorphic function. This result is related to the study of compact Jacobi matrices. We indicate how this notion was developped and used since Stieltjes, with special attention to the results by M. G. Krein. We also pay attention to the perturbation of a constant Jacobi matrix by a compact Jacobi matrix, work which basically started with Blumenthal in 1889 and which now is known as the theory for the class M(a,b)M(a,b)

    Zero distribution of polynomials satisfying a differential-difference equation

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    In this paper we investigate the asymptotic distribution of the zeros of polynomials Pn(x)P_{n}(x) satisfying a first order differential-difference equation. We give several examples of orthogonal and non-orthogonal families.Comment: 26 pages, 2 figure

    Orthogonal polynomials and Laurent polynomials related to the Hahn-Exton q-Bessel function

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    Laurent polynomials related to the Hahn-Exton qq-Bessel function, which are qq-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurent qq-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orthogonal polynomials the orthogonality measure is determined using the three-term recurrence relation as a starting point. The relation between Chebyshev polynomials of the second kind and the Laurent qq-Lommel polynomials and related functions is used to obtain estimates for the latter
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