Shannon entropy of symmetric Pollaczek polynomials

Martínez–Finkelshtein, Andrei; Sánchez-Lara, J.

doi:10.1016/j.jat.2006.06.007

Cited by 4 publications

(2 citation statements)

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“…where c is a positive constant; see [21,[393][394][395][396][397][398][399][400] for a detailed description. We should like to mention, as a brief guide to the reader, some recent literature on the Pollaczek polynomials; see [22] for the asymptotic behavior of the polynomials for x / ∈ [−1, 1]; [2,24] for the asymptotic behavior of their zeros and [14,15,23] for applications to physical problems. Regarding weighted polynomial approximation in L p with respect to a class of exponential weights on [−1, 1] that violate the Szegö conditions, see [9,13].…”

Section: Introductionmentioning

confidence: 99%

Painlevé V and a Pollaczek–Jacobi type orthogonal polynomials

Chen

Dai

2010

Journal of Approximation Theory

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We study a sequence of polynomials orthogonal with respect to a one-parameter family of weightsdefined for x ∈ [0, 1]. If t = 0, this reduces to a shifted Jacobi weight. Our ladder operator formalism and the associated compatibility conditions give an easy determination of the recurrence coefficients.For t > 0, the factor e −t/x induces an infinitely strong zero at x = 0. With the aid of the compatibility conditions, the recurrence coefficients are expressed in terms of a set of auxiliary quantities that satisfy a system of difference equations. These, when suitably combined with a pair of Toda-like equations derived from the orthogonality principle, show that the auxiliary quantities are particular Painlevé V and/or allied functions.It is also shown that the logarithmic derivative of the Hankel determinant,, satisfies the Jimbo-Miwa-Okamoto σ -form of the Painlevé V equation and that the same quantity satisfies a second-order non-linear difference equation which we believe to be new. Crown

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Section: Introductionmentioning

confidence: 99%

Painlevé V and a Pollaczek–Jacobi type orthogonal polynomials

Chen

Dai

2010

Journal of Approximation Theory

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“…; see e.g. [3,4,7,10,11,12,13,20] and the references therein. In particular, it has been shown that for Chebyshev orthonormal polynomials of the first kind the relative entropy K n does not depend on n,…”

Section: Introductionmentioning

confidence: 99%