Generalized Jacobi orthogonal polynomials

Bouras, B.

doi:10.1080/10652460701445666

Cited by 7 publications

(3 citation statements)

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“…It is simple to find dµ(z) = e The reason for doing this change of variables is because orthogonal polynomials with respect to this last measure, (4.57), are known. In fact, in the same way that in the single Penner potential (which is k = 1 in (4.46)) we deal with the Laguerre polynomials (see, e.g., [11]) in here the relevant orthogonal polynomials are the generalized Gegenbauer polynomials [77,78,79]. Their precise definition is In order to compute the partition function (2.13) the next step is to address the coefficients h n .…”

Section: The Triple Penner Potential and Agt Stokes Phenomenamentioning

confidence: 99%

The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painlevé II Equation

Schiappa

Vaz

2014

Commun. Math. Phys.

138

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Resurgent transseries have recently been shown to be a very powerful construction in order to completely describe nonperturbative phenomena in both matrix models and topological or minimal strings. These solutions encode the full nonperturbative content of a given gauge or string theory, where resurgence relates every (generalized) multi-instanton sector to each other via large-order analysis. The Stokes phase is the adequate gauge theory phase where an 't Hooft large N expansion exists and where resurgent transseries are most simply constructed. This paper addresses the nonperturbative study of Stokes phases associated to multi-cut solutions of generic matrix models, constructing nonperturbative solutions for their free energies and exploring the asymptotic large-order behavior around distinct multi-instanton sectors. Explicit formulae are presented for the Z 2 symmetric two-cut set-up, addressing the cases of the quartic matrix model in its two-cut Stokes phase; the "triple" Penner potential which yields four-point correlation functions in the AGT framework; and the Painlevé II equation describing minimal superstrings.

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Section: The Triple Penner Potential and Agt Stokes Phenomenamentioning

confidence: 99%

The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painlevé II Equation

Schiappa

Vaz

2014

Commun. Math. Phys.

138

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“…In collaboration with J. Alaya, the first author started in Refs [2,3] with some families of symmetric D-semiclassical MOPS {ß«}"ao of class . y < 2 and they proved that the sequences {P"}"sio in their quadratic decomposition (1.3) are D-semiclassical of class one.…”

Section: B Bouras and F Marcenanmentioning

confidence: 99%

Quadratic decomposition of a family ofH_q-semiclassical orthogonal polynomial sequences

Bouras

Marcellán

2012

Journal of Difference Equations and Applications

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We deal with monic orthogonal polynomial sequences (MOPSs), {ß,,},,^,,, satisfying the three-term recurrence relation 5,,+2(x) = (x -;S"^.|)ß,,+i(x) -7"+|ß"(x), n = O, 1,2, ..., with initial conditions B^ix) = 1 and S|(jr) = x -ßo, where ß" = (-l)"jßo and y,, ¥" O for all n & 1. These sequences are characterized by the relation ß2"(x) = P"ix^), n > O, ß|(x) = A--A), where {P,,),,ao is a MOPS. In this paper, we show that the sequence {S,,],,20 is W^-semiclassical if and only if the sequence {P,,},,^() is //^2-semiclassical. Then, we express the characteristic elements of the //(^-.semiclassical sequence (fi,, },,2(,, .such as the ç-Pearson equation satisfied by the corresponding linear functional, the class ofthe linear functional, the first-order linear q-difference equation satisfied by the Stieltjes function and the coefficients of the structure relation for such a sequence of polynomials, in terms of the characteristic elements of the sequence, {f,,},,a.(). In particular, if the .sequence {i*,,},,=;() is //^2-semiclassical of class zero, then we obtain a new non-symmetric //,-semiclassical sequence of polynomials \B" },,>o of class .5=1.

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“…A large number of relevant properties of the Jacobi orthogonal polynomials and its applications are available e.g. [1], [2], [3], [4], [5], [7], [8], [9], and [10].…”

Section: Introductionmentioning

confidence: 99%

Extended Jacobi polynomials

Amir¹

2014

ijcms

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Generalized Jacobi orthogonal polynomials

Cited by 7 publications

References 12 publications

The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painlevé II Equation

The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painlevé II Equation

Quadratic decomposition of a family ofH_q-semiclassical orthogonal polynomial sequences

Extended Jacobi polynomials

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Generalized Jacobi orthogonal polynomials

Cited by 7 publications

References 12 publications

The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painlevé II Equation

The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painlevé II Equation

Quadratic decomposition of a family ofHq-semiclassical orthogonal polynomial sequences

Extended Jacobi polynomials

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Quadratic decomposition of a family ofH_q-semiclassical orthogonal polynomial sequences