On semiclassical linear functionals: integral representations

Marcellán, Francisco; Rocha, I. A.

doi:10.1016/0377-0427(93)e0248-k

Cited by 21 publications

(30 citation statements)

References 4 publications

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“…The weights we are considering are all of the semiclassical type as defined in [10,12,13,2,4]. This means that they are of the form µ(x) = e −V (x) with V (x) an arbitrary rational function.…”

Section: Hankel and Shifted Töplitz Determinantsmentioning

confidence: 99%

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The Dependence on the Monodromy Data of the Isomonodromic Tau Function

Bertola

2009

Commun. Math. Phys.

132

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The isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the Malgrange's divisor of generalized monodromy data for which a vector bundle is nontrivial, or, which is the same, a certain Riemann-Hilbert problem has no solution. In their original work, Jimbo, Miwa, Ueno provided an algebraic construction of its derivatives with respect to isomonodromic times. However the dependence on the (generalized) monodromy data (i.e. monodromy representation and Stokes' parameters) was not derived. We fill the gap by providing a (simpler and more general) description in which all the parameters of the problem (monodromy-changing and monodromy-preserving) are dealt with at the same level. We thus provide variational formulae for the isomonodromic tau function with respect to the (generalized) monodromy data. The construction applies more generally: given any (sufficiently well-behaved) family of Riemann-Hilbert problems (RHP) where the jump matrices depend arbitrarily on deformation parameters, we can construct a one-form Ω (not necessarily closed) on the deformation space (Malgrange's differential), defined off Malgrange's divisor. We then introduce the notion of discrete Schlesinger transformation: it means that we allow the solution of the RHP to have poles (or zeros) at prescribed point(s). Even if Ω is not closed, its difference evaluated along the original solution and the transformed one, is shown to be the logarithmic differential (on the deformation space) of a function. As a function of the position of the points of the Schlesinger transformation, yields a natural generalization of Sato formula for the Baker-Akhiezer vector even in the absence of a tau function, and it realizes the solution of the RHP as such BA vector.Some exemplifications in the setting of the Painlevé II equation and finite Töplitz/Hankel determinants are provided.

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Section: Hankel and Shifted Töplitz Determinantsmentioning

confidence: 99%

“…The notion of semiclassical symbols was defined in [12] and it was shown in [1] that the corresponding Hankel/Toeplitz determinant were isomonodromic tau function, to within an explicit non-vanishing factor. The weights we are considering are all of the semiclassical type as defined in [10,12,13,2,4].…”

Section: Hankel and Shifted Töplitz Determinantsmentioning

confidence: 99%

The Dependence on the Monodromy Data of the Isomonodromic Tau Function

Bertola

2009

Commun. Math. Phys.

132

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“…This class of linear functionals is sometimes referred to as semiclassical moment functionals [3,14,15]. We consider the corresponding monic generalized orthogonal polynomials p n (x), which satisfy…”

Section: Orthogonality Measures and Integration Contoursmentioning

confidence: 99%

“…The deformation equations (2-72), (2-73) (2-77) are equivalent to the infinite sequence of 2 × 2 equations (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) where the folded matrix of the deformation is defined by…”

Section: Proposition 32mentioning

confidence: 99%

“…Remark 3.1 Note that formula (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) for the deformation of the measure in Proposition 3.2, as well as those below, (3-36), (3-37), which are obtained through folding of the ∂x operator, could also be derived for arbitrary locally analytic potentials V (x), provided all the integrals involved are convergent [13]. However applicability of the subsequent isomonodromic analysis would be lost if the derivatives were not rational, since the resulting deformation equations would then have essential singularities.…”

Section: Proposition 32mentioning

confidence: 99%

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Semiclassical Orthogonal Polynomials, Matrix Models and Isomonodromic Tau Functions

Bertola

Eynard

Harnad

2006

Commun. Math. Phys.

122

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The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such measures. These are shown to preserve the generalized monodromy of the associated rank-2 rational covariant derivative operators. The corresponding matrix models, consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed, and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or, more generally, the gap probablities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions.

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Classification of Laguerre–Hahn orthogonal polynomials of class one

Filipuk

Rebocho

2019

Mathematische Nachrichten

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We study orthogonal polynomials related to Stieltjes functions satisfying Riccati type differential equations with polynomial coefficients,We derive recurrences for the three-term recurrence relation coefficients of the orthogonal polynomials, including connections with some forms of discrete Painlevé equations. K E Y W O R D SLaguerre-Freud equations, orthogonal polynomials, Painlevé equations, Stieltjes function M S C ( 2 0 1 0 ) 33C45, 33C47, 42C05 MOTIVATIONOrthogonal polynomials { ( ) = + lower degree terms} ≥0 on the real line may be fully characterized by the orthogonality relation1) or, equivalently, by a three-term recurrence relation [27] +1 ( ) = ( − ) ( ) − −1 ( ), = 1, 2, … , (1.2) with 0 ( ) = 1, 1 ( ) = − 0 and ≠ 0, ≥ 1. Here, , is the Kronecker delta, and is a positive Borel measure supported on a finite or infinite interval of the real line, , with finite moments 244

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On semiclassical linear functionals: integral representations

Cited by 21 publications

References 4 publications

The Dependence on the Monodromy Data of the Isomonodromic Tau Function

The Dependence on the Monodromy Data of the Isomonodromic Tau Function

Semiclassical Orthogonal Polynomials, Matrix Models and Isomonodromic Tau Functions

Classification of Laguerre–Hahn orthogonal polynomials of class one

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