A new quantum algebraic interpretation of the Askey-Wilson polynomials

AbstractWe express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulæ for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.

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Befriending Askey–Wilson polynomials

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025714500155 ◽

2014 ◽

Vol 17 (03) ◽

pp. 1450015 ◽

Cited By ~ 4

Author(s):

Paweł J. Szabłowski

Keyword(s):

Characteristic Function ◽

Hermite Polynomials ◽

Probabilistic Interpretation ◽

Conjugate Pair ◽

Linear Combinations ◽

Expansion Formula ◽

Wilson Polynomials ◽

The Way

We recall five families of polynomials constituting a part of the so-called Askey–Wilson scheme. We do this to expose properties of the Askey–Wilson (AW) polynomials that constitute the last, most complicated element of this scheme. In doing so we express AW density as a product of the density that makes q-Hermite polynomials orthogonal times a product of four characteristic function of q-Hermite polynomials (2.9) just pawing the way to a generalization of AW integral. Our main results concentrate mostly on the complex parameters case forming conjugate pairs. We present new fascinating symmetries between the variables and some newly defined (by the appropriate conjugate pair) parameters. In particular in (3.12) we generalize substantially famous Poisson–Mehler expansion formula (3.16) in which q-Hermite polynomials are replaced by Al-Salam–Chihara polynomials. Further we express Askey–Wilson polynomials as linear combinations of Al-Salam–Chihara (ASC) polynomials. As a by-product we get useful identities involving ASC polynomials. Finally by certain re-scaling of variables and parameters we reach AW polynomials and AW densities that have clear probabilistic interpretation.

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Expansions in the Askey–Wilson polynomials

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2014.11.048 ◽

2015 ◽

Vol 424 (1) ◽

pp. 664-674 ◽

Cited By ~ 3

Author(s):

Mourad E.H. Ismail ◽

Dennis Stanton

Keyword(s):

Wilson Polynomials

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Lie-algebraic interpretation of the maximal superintegrability and exact solvability of the Coulomb–Rosochatius potential inndimensions

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8113/44/35/355208 ◽

2011 ◽

Vol 44 (35) ◽

pp. 355208 ◽

Cited By ~ 1

Author(s):

G A Kerimov ◽

A Ventura

Keyword(s):

Exact Solvability ◽

Algebraic Interpretation

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The factorization method for the Askey–Wilson polynomials

Journal of Computational and Applied Mathematics ◽

10.1016/s0377-0427(99)00090-4 ◽

1999 ◽

Vol 107 (2) ◽

pp. 219-232 ◽

Cited By ~ 13

Author(s):

Gaspard Bangerezako

Keyword(s):

Factorization Method ◽

Wilson Polynomials

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Multilayer networks as embodied consciousness interactions. A formal model approach.

10.31234/osf.io/3y8at ◽

2021 ◽

Author(s):

Camilo Miguel Signorelli ◽

joaquin diaz boils

Keyword(s):

Formal Model ◽

Conscious Experience ◽

Formal Analysis ◽

Mathematical Formulation ◽

Multilayer Networks ◽

Topological Network ◽

Binary Operator ◽

Algebraic Interpretation ◽

Embodied Consciousness ◽

Model Approach

An algebraic interpretation of multilayer networks is introduced in relation to conscious experience, brain and body. The discussion is based on a network model for undirected multigraphs with coloured edges whose elements are time-evolving multilayers, representing complex experiential brain-body networks. These layers have the ability to merge by an associative binary operator, accounting for biological composition. As an extension, they can rotate in a formal analogy to how the activity inside layers would dynamically evolve. Under consciousness interpretation, we also studied a mathematical formulation of splitting layers, resulting in a formal analysis for the transition from conscious to non-conscious activity. From this construction, we recover core structures for conscious experience, dynamical content and causal efficacy of conscious interactions, predicting topological network changes after conscious layer interactions. Our approach provides a mathematical account of coupling and splitting layers co-arising with more complex experiences. These concrete results may inspire the use of formal studies of conscious experience not only to describe it, but also to obtain new predictions and future applications of formal mathematical tools.

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