Asymptotic distributions of the zeros of a family of hypergeometric polynomials

Zhou, Jian; Srivástava, H. M.; Wang, Zhigang

doi:10.1090/s0002-9939-2011-11117-1

Cited by 13 publications

(10 citation statements)

References 20 publications

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“…If k is even there is always a real saddle, whereas if it is odd all saddles are complex. As k grows, the saddles asymptote to the unit circle from inside its disk [181]. These properties are illustrated in figure 2.…”

Section: Spectral Geometries and Instanton Actionsmentioning

confidence: 90%

From Minimal Strings towards Jackiw-Teitelboim Gravity: On their Resurgence, Resonance, and Black Holes

Gregori,

Schiappa

2021

Preprint

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Two remarkable facts about Jackiw-Teitelboim two-dimensional dilaton-gravity have been recently uncovered: this theory is dual to an ensemble of quantum mechanical theories; and such ensemble is described by a random matrix model which itself may be regarded as a special (large matter-central-charge) limit of minimal string theory. This work addresses this limit, putting it in its broader matrix-model context; comparing results between multicritical models and minimal strings (i.e., changing in-between multicritical and conformal backgrounds); and in both cases making the limit of large matter-central-charge precise (as such limit can also be defined for the multicritical series). These analyses are first done via spectral geometry, at both perturbative and nonperturbative levels, addressing the resurgent large-order growth of perturbation theory, alongside a calculation of nonperturbative instanton-actions and corresponding Stokes data. This calculation requires an algorithm to reach large-order, which is valid for arbitrary two-dimensional topological gravity. String equations-as derived from the Gel'fand-Dikii construction of the resolvent-are analyzed in both multicritical and minimal string theoretic contexts, and studied both perturbatively and nonperturbatively (always matching against the earlier spectral-geometry computations). The resulting solutions, as described by resurgent transseries, are shown to be resonant. The large matter-central-charge limit is addressed-in the string-equation context-and, in particular, the string equation for Jackiw-Teitelboim gravity is obtained to next derivative-orders, beyond the known genus-zero case (its possible exact-form is also discussed). Finally, a discussion of gravitational perturbations to Schwarzschild-like black hole solutions in these minimal-string models, regarded as deformations of Jackiw-Teitelboim gravity, is included-alongside a brief discussion of quasinormal modes.

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Section: Spectral Geometries and Instanton Actionsmentioning

confidence: 90%

From Minimal Strings towards Jackiw-Teitelboim Gravity: On their Resurgence, Resonance, and Black Holes

Gregori,

Schiappa

2021

Preprint

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“…There exist several well-known and important families of hypergeometric polynomials [5,21]. What they all have in common is the fact that they satisfy certain linear differential equations with polynomial coefficients that are special instances of (2.3).…”

Section: Hypergeometric Polynomials In Several Variablesmentioning

confidence: 99%

“…The univariate case has been extensively studied both classically (see e.g. [10,12]) and recently (see [3,4,21] and the references therein). Already the distribution of zeros of polynomial instances of the simplest non-elementary hypergeometric function 2 F 1 (a, b; c; x) is far from being clear.…”

Section: Introductionmentioning

confidence: 99%

Hypergeometric polynomials are optimal

Bogdanov

Sadykov

2019

Math. Z.

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With any integer convex polytope P ⊂ R n we associate a multivariate hypergeometric polynomial whose set of exponents is Z n ∩ P. This polynomial is defined uniquely up to a constant multiple and satisfies a holonomic system of partial differential equations of Horn's type. We prove that under certain nondegeneracy conditions the zero locus of any such polynomial is optimal in the sense of [7], i.e., that the topology of its amoeba [11] is as complex as it could possibly be. Using this, we derive optimal properties of several classical families of multivariate hypergeometric polynomials.

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“…It is known [16] that the zeros of P n approach the unit circle as n tends to infinity. In addition to that, we are going to show that in the limit, the zeros are distributed uniformly on the unit circle.…”

Section: Sums Of Squared Baskakov Functionsmentioning

confidence: 99%

“…Proof. We begin by recapitulating the asymptotic behavior of P n from the proof of Theorem 1 in [16]. For every 0 < ǫ < 1 we have…”

Section: Sums Of Squared Baskakov Functionsmentioning

confidence: 99%

Complete monotonicity and zeros of sums of squared Baskakov functions

Abel

Gawronski

Neuschel

2015

Applied Mathematics and Computation

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Keywords:Baskakov operator Complete monotonicity Convexity Chebyshev-Grüss-type inequality Distribution of zeros Complete elliptic integral of the first kind a b s t r a c tWe prove complete monotonicity of sums of squares of generalized Baskakov basis functions by deriving the corresponding results for hypergeometric functions. Moreover, in the central Baskakov case we study the distribution of the complex zeros for large values of a parameter. We finally discuss the extension of some results for sums of higher powers.

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Asymptotic distributions of the zeros of a family of hypergeometric polynomials

Cited by 13 publications

References 20 publications

From Minimal Strings towards Jackiw-Teitelboim Gravity: On their Resurgence, Resonance, and Black Holes

From Minimal Strings towards Jackiw-Teitelboim Gravity: On their Resurgence, Resonance, and Black Holes

Hypergeometric polynomials are optimal

Complete monotonicity and zeros of sums of squared Baskakov functions

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