1,011 research outputs found

    Towards a full solution of the large N double-scaled SYK model

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    We compute the exact, all energy scale, 4-point function of the large NN double-scaled SYK model, by using only combinatorial tools and relating the correlation functions to sums over chord diagrams. We apply the result to obtain corrections to the maximal Lyapunov exponent at low temperatures. We present the rules for the non-perturbative diagrammatic description of correlation functions of the entire model. The latter indicate that the model can be solved by a reduction of a quantum deformation of SL(2)(2), that generalizes the Schwarzian to the complete range of energies.Comment: 52+28 pages, 14 figures; v2: references revised, typos corrected, changed normalization of SL(2)_q 6j symbo

    Difference Sturm--Liouville problems in the imaginary direction

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    We consider difference operators in L2L^2 on R\R of the form Lf(s)=p(s)f(s+i)+q(s)f(s)+r(s)f(si), L f(s)=p(s)f(s+i)+q(s) f(s)+r(s) f(s-i) , where ii is the imaginary unit. The domain of definiteness are functions holomorphic in a strip with some conditions of decreasing at infinity. Problems of such type with discrete spectra are well known (Meixner--Pollaszek, continuous Hahn, continuous dual Hahn, and Wilson hypergeometric orthogonal polynomials). We write explicit spectral decompositions for several operators LL with continuous spectra. We also discuss analogs of 'boundary conditions' for such operators.Comment: 27p

    Symmetry Groups of AnA_n Hypergeometric Series

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    Structures of symmetries of transformations for Holman-Biedenharn-Louck AnA_n hypergeometric series: AnA_n terminating balanced 4F3{}_4 F_3 series and AnA_n elliptic 10E9{}_{10} E_9 series are discussed. Namely the description of the invariance groups and the classification all of possible transformations for each types of AnA_n hypergeometric series are given. Among them, a "periodic" affine Coxeter group which seems to be new in the literature arises as an invariance group for a class of AnA_n 4F3{}_4 F_3 series

    Jacobian elliptic Kummer surfaces and special function identities

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    We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities relating differential systems are obtained. They include a new identity relating Appell's hypergeometric system to a product of two Gauss' hypergeometric differential equations by a cubic transformation.Comment: 20 page

    Dispersion and limit theorems for random walks associated with hypergeometric functions of type BC

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    The spherical functions of the noncompact Grassmann manifolds Gp,q(F)=G/KG_{p,q}(\mathbb F)=G/K over the (skew-)fields F=R,C,H\mathbb F=\mathbb R, \mathbb C, \mathbb H with rank q1q\ge1 and dimension parameter p>qp>q can be described as Heckman-Opdam hypergeometric functions of type BC, where the double coset space G//KG//K is identified with the Weyl chamber CqBRq C_q^B\subset \mathbb R^q of type B. The corresponding product formulas and Harish-Chandra integral representations were recently written down by M. R\"osler and the author in an explicit way such that both formulas can be extended analytically to all real parameters p[2q1,[p\in[2q-1,\infty[, and that associated commutative convolution structures p*_p on CqBC_q^B exist. In this paper we introduce moment functions and the dispersion of probability measures on CqBC_q^B depending on p*_p and study these functions with the aid of this generalized integral representation. Moreover, we derive strong laws of large numbers and central limit theorems for associated time-homogeneous random walks on (CqB,p)(C_q^B, *_p) where the moment functions and the dispersion appear in order to determine drift vectors and covariance matrices of these limit laws explicitely. For integers pp, all results have interpretations for GG-invariant random walks on the Grassmannians G/KG/K. Besides the BC-cases we also study the spaces GL(q,F)/U(q,F)GL(q,\mathbb F)/U(q,\mathbb F), which are related to Weyl chambers of type A, and for which corresponding results hold. For the rank-one-case q=1q=1, the results of this paper are well-known in the context of Jacobi-type hypergroups on [0,[[0,\infty[.Comment: Extended version of arXiv:1205.4866; some corrections to prior version. Accepted for publication in J. Theor. Proba

    On the distribution of exponential functionals for Levy processes with jumps of rational transform

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    We derive explicit formulas for the Mellin transform and the distribution of the exponential functional for Levy processes with rational Laplace exponent. This extends recent results by Cai and Kou on the processes with hyper-exponential jumps [N. Cai and S. Kou "Prising Asian options under a general jump diffusion model", (2010)].Comment: 11 page

    Distinct parts partitions without sequences

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    Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties, hypergeometric representations for the generating functions, and asymptotic formulas for the enumeration functions. We complete a similar investigation of partitions into distinct parts without sequences, which are of particular interest due to their relationship with the Rogers-Ramanujan identities. Our main results include a double series representation for the generating function, an asymptotic formula for the enumeration function, and several combinatorial inequalities.Comment: 15 page
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