Multivariate Jacobi polynomials and the Selberg integral

Olshanski, Grigori; Osinenko, A.

doi:10.1007/s10688-012-0034-0

Cited by 13 publications

(25 citation statements)

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“…was proved in [27] for all z, z ′ ∈ {u ∈ C : ℜu > − (1+b) 2 } and extended by analytic continuation, in [13], for all pairs (z, z ′ ) in the complex domain…”

Section: Bc Type Z-measuresmentioning

confidence: 97%

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BC type z-measures and determinantal point processes

Cuenca

2018

Advances in Mathematics

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The (BC type) z-measures are a family of four parameter z, z ′ , a, b probability measures on the path space of the nonnegative Gelfand-Tsetlin graph with Jacobi-edge multiplicities. We can interpret the z-measures as random point processes P z,z ′ ,a,b on the punctured positive real line X = R >0 \ {1}. Our main result is that these random processes are determinantal and moreover we compute their correlation kernels explicitly in terms of hypergeometric functions.For very special values of the parameters z, z ′ , the processes P z,z ′ ,a,b on X are essentially scaling limits of Racah orthogonal polynomial ensembles and their correlation kernels can be computed simply from some limits of the Racah polynomials. Thus, in the language of random matrices, we study certain analytic continuations of processes that are limits of Racah ensembles, and such that they retain the determinantal structure. Another interpretation of our results, and the main motivation of this paper, is the representation theory of big groups. In representation-theoretic terms, this paper solves a natural problem of harmonic analysis for several infinite-dimensional symmetric spaces.

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“…was proved in [27] for all z, z ′ ∈ {u ∈ C : ℜu > − (1+b) 2 } and extended by analytic continuation, in [13], for all pairs (z, z ′ ) in the complex domain…”

Section: Bc Type Z-measuresmentioning

confidence: 97%

“…which implies ǫ ≥ 0. This restriction allows us to use the main results from [24,27]. • In Section 2 below, we define several subsets of C 2 from which we take pairs (z, z ′ ) as parameters of the z-measures and for some proofs involving analytic continuations.…”

Section: Notation and Terminologymentioning

confidence: 99%

BC type z-measures and determinantal point processes

Cuenca

2018

Advances in Mathematics

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“…However, in parts of the paper, we could assume the less restrictive a, b > −1, which is the only requirement necessary for the existence of the Jacobi polynomials P λ (·|a, b). The assumption a ≥ b ≥ −1/2 is required in order to make use of the results from [24,30]. We keep the stronger assumption throughout to make estimates a little easier.…”

Section: Conventionsmentioning

confidence: 99%

“…The calculation of all three terms in (A.3) is carried out in [30], for general parameters a, b > −1 and z ∈ C, ℜz > −(1 + b)/2, and the answer is given by the formula (1.3) for z ′ = z. The expression (1.3), with representation-theoretic origin for z ′ = z, and the pairs (a, b) ∈ {(0, 0), (1/2, 1/2), (±1/2, −1/2)}, suggests to consider an analytic continuation replacing z by z ′ and as general real parameters a, b as possible.…”

Section: Appendix a Z-measures And Harmonic Analysis On Big Groupsmentioning

confidence: 99%

Markov processes on the duals to infinite-dimensional classical Lie groups

Cuenca¹

2018

Ann. Inst. H. Poincaré Probab. Statist.

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“…Analogues of the zw-measures also exist for other infinitedimensional classical groups and for infinite-dimensional symmetric spaces (see Olshanski and Osinenko's paper [14]). The zw-measures play a fundamental role in infinite-dimensional harmonic analysis, because their scaling limits govern the spectral decomposition of certain distinguished unitary representations.…”

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confidence: 95%

Determinantal measures related to big q-Jacobi polynomials

Gorin

Olshanski

2015

Funct Anal Its Appl

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Abstract. We define a novel combinatorial object-the extended Gelfand-Tsetlin graph with cotransition probabilities depending on a parameter q. The boundary of this graph admits an explicit description. We introduce a family of probability measures on the boundary and describe their correlation functions. These measures are a q-analogue of the spectral measures studied earlier in the context of the problem of harmonic analysis on the infinite-dimensional unitary group.Key words: Gelfand-Tsetlin graph, determinantal measures, big q-Jacobi polynomials, basic hypergeometric series.Determinantal measures form a special class of probability measures on spaces of locally finite point configurations (see Borodin's survey [1]). The key property of a determinantal measure is that its correlation functions of any order are expressed in a simple way through a function in two variables, called the correlation kernel.A family of determinantal measures, called the zw-measures, was studied by Borodin and Olshanski [2] in connection with the problem of harmonic analysis on the infinite-dimensional unitary group U (∞), posed by Olshanski [12]. Analogues of the zw-measures also exist for other infinitedimensional classical groups and for infinite-dimensional symmetric spaces (see Olshanski and Osinenko's paper [14]). The zw-measures play a fundamental role in infinite-dimensional harmonic analysis, because their scaling limits govern the spectral decomposition of certain distinguished unitary representations.On the other hand, the zw-measures are nice combinatorial objects and have much in common with the so-called z-measures, which are a particular case of Okounkov's Schur measures on partitions. Like Schur measures, the zw-measures admit a generalization involving an additional deformation parameter similar to Dyson's β-parameter in random matrix theory or the continuous parameter of the Jack symmetric functions (see Olshanski's paper [13]).Our aim is to show that the notion of zw-measures can be extended in another direction; namely, there exists a (nonevident) q-analogue of the zw-measures. Our first result says that the "N -particle q-zw-measures" have a large-N limit, and the limiting probability measure is a determinantal measure on a space of infinite point configurations. We find the corresponding correlation kernelit is expressed through the basic hypergeometric function 2 φ 1 . The large-N limit transition is related to another result, which is of independent interest-a description of the q-boundary for an extended version of the Gelfand-Tsetlin graph.1. The q-analogue of zw-measures. We fix a triple (ζ + , ζ − , q) of real parameters, where ζ + > 0, ζ − < 0, and 0 < q < 1. The corresponding double q-lattice is a subsetBy a configuration we mean an arbitrary subset X ⊂ L. For N = 1, 2, . . . , let G N denote the countable set consisting of all N -point configurations. We enumerate the points of every X ∈ G N in increasing order and write X = (x 1 < · · · < x N ).We are going to introduce, for every N = 1, 2, . . . , a ...

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Multivariate Jacobi polynomials and the Selberg integral

Cited by 13 publications

References 23 publications

BC type z-measures and determinantal point processes

BC type z-measures and determinantal point processes

Markov processes on the duals to infinite-dimensional classical Lie groups

Determinantal measures related to big q-Jacobi polynomials

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