Jyoichi Kaneko
We prove a generalization of the $q$-Selberg integral evaluation formula. The integrand is that of $q$-Selberg integral multiplied by a factor of the same form with respect to part of the variables. The proof relies on the quadratic norm formula of Koornwinder polynomials. We also derive generalizations of Mehta's integral formula as limit cases of our integral.
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We construct new families of (q-) difference and (contour) integral operators having nice actions on Koornwinder's multivariate orthogonal polynomials. We further show that the Koornwinder polynomials can be constructed by suitable sequences of these operators applied to the constant polynomial 1, giving the difference-integral representation of the title. Macdonald's conjectures (as proved by van Diejen and Sahi) for the principal specialization and norm follow immediately, as does a Cauchy-type identity of Mimachi.
The classical Selberg integral contains a power of the Vandermonde determinant. When that power is a square, it is easy to prove Selberg's identity by interpreting it as a determinant of one-variable integrals. We give similar proofs of summation and transformation formulas for continuous and discrete elliptic Selberg integrals. In the continuous case, the same proof was previously given by Noumi. Special cases of these identities have found applications in combinatorics.
In a recent paper Richards and Zheng compute the determinant of a matrix whose entries are given by beta-type integrals, thereby generalising an earlier result by Dixon and Varchenko. They then use their result to obtain a generalisation of the famous Selberg integral. In this note we point out that the Selberg-generalisation of Richards and Zheng is a special case of an integral over Jack polynomials due to Kadell. We then show how an integral formula for Jack polynomials of Okounkov and Olshanski may be applied to prove Kadell's integral along the lines of Richards and Zheng.
In this paper, we describe a general method for computing Selberg-like integrals based on a formula, due to Kaneko, for Selberg-Jack integrals. The general principle consists in expanding the integrand \emph{w.r.t.} the Jack basis, which is obtained by a Gram-Schmidt orthogonalization process. The resulting algorithm is not very efficient because of this decomposition. But for special cases, the coefficients admit a closed form. As an example, we study the case of the power-sums since for which the coefficients are obtained by manipulating generating series by means of transformations of alpha…
We obtain an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole. The complexity of our expression depends on the distance from the hole to the center of the hexagon. This proves and generalizes conjectures of Ciucu et al., who considered the case of plain enumeration when the triangle is located at or very near the center. Our proof uses Askey-Wilson polynomials as a tool to relate discrete and continuous Selberg-type integrals.
Using an extension of the well-known evaluation symmetry, a new Cauchy-type identity for Macdonald polynomials is proved. After taking the classical limit this yields a new sl_3 generalisation of the famous Selberg integral. Closely related results obtained in this paper are an sl_3-analogue of the Askey-Habsieger-Kadell q-Selberg integral and an extension of the q-Selberg integral to a transformation between q-integrals of different dimensions.
A new q-binomial theorem for Macdonald polynomials is employed to prove an A_n analogue of the celebrated Selberg integral. This confirms the g=A_n case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra g.
Elliptic Selberg integrals and conformal blocks
本文给出了Selberg积分公式的一个椭圆形式。
Elliptic Selberg integrals
本文给出了Selberg积分公式的一个椭圆形式。
We show how to determine the asymptotics of a certain Selberg-type integral by means of tools available in the theory of (generalised) hypergeometric series. This provides an alternative derivation of a result of Carré, Deneufchâtel, Luque and Vivo [arXiv:1003.5996].
It has been remarked that a fair measure of the impact of Atle Selberg's work is the number of mathematical terms which bear his name. One of these is the Selberg integral, an n-dimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after publication. In quick succession the Selberg integral was used to prove an outstanding conjecture in random matrix theory, and cases of the Macdonald conjectures. It further initiated the study of q-analogues, which in turn enriched the Macd…
In their 2011 paper on the AGT conjecture, Alba, Fateev, Litvinov and Tarnopolsky (AFLT) obtained a closed-form evaluation for a Selberg integral over the product of two Jack polynomials, thereby unifying the well-known Kadell and Hua--Kadell integrals. In this paper we use a variety of symmetric functions and symmetric function techniques to prove generalisations of the AFLT integral. These include (i) an $\mathrm{A}_n$ analogue of the AFLT integral, containing two Jack polynomials in the integrand; (ii) a generalisation of (i) for $γ=1$ (the Schur or GUE case), containing a product of $n+1$ …
In this paper, we present two new representations of the alternating Zeta function. We show that for any s $\in$ C this function can be computed as a limit of a series of determinant. We then express these determinants as the expectation of a functional of a random vector with Dixon-Anderson density. The generalization of this representation to more general alternating series allows us to evaluate a Selberg-type integral with a generalized Vandermonde determinant.
We provide an explicit expression for the first order $q$-difference system for the Jackson integral of symmetric Selberg type. The $q$-difference system gives a generalization of $q$-analog of contiguous relations for the Gauss hypergeometric function. As a basis of the system we use a set of the symmetric polynomials introduced by Matsuo in his study of the $q$-KZ equation. Our main result is an explicit expression for the coefficient matrix of the $q$-difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called interpolation polynomials,…
In the previous paper (J. Combin. Theory Ser. A, 120, 2013, 1263--1284) H. Tagawa and the two authors proposed an algebraic method to compute certain Pfaffians whose form resemble to Hankel determinants associated with moment sequences of the classical orthogonal polynomials. At the end of the paper they offered several conjectures. In this work we employ a completely different method to evaluate this type of Pfaffians. The idea is to apply certain de Bruijn type formula and convert the evaluation of the Pfaffians to the certain Selberg type integrals. This approach works not only for Pfaffian…
The connection problem associated with a Selberg type integral is solved. The connection coefficients are given in terms of the $q$-Racah polynomials. As an application of the explicit expression of the connection coefficients, examples of the monodromy-invariant Hermitian form of non-diagonal type are presented. It is noteworthy that such Hermitian forms are intimately related with the correlation functions of non-diagonal type in $\hat{sl_2}$-confromal field theory.
The singularities of Selberg- and Dotsenko-Fateev-like integrals
本文通过几何分析被积函数奇点，研究Selberg型与Dotsenko-Fateev型超几何积分的亚纯延拓，并证明对称或DF对称情形下的多个表观奇点可移除。
Elliptic $\mathrm{A}_n$ Selberg integrals
本文借助第二作者的椭圆插值核，证明椭圆 Selberg 积分的 $\mathrm{A}_n$ 推广，并进一步得到 $\mathrm{A}_n$ Kadell、Hua-Kadell 与 AFLT 积分的椭圆类似物。
We study $β$-Jacobi diffusion processes on alcoves in $\mathbb R^N$, depending on 3 parameters. Using elementary symmetric functions, we present space-time-harmonic functions and martingales for these processes $(X_t)_{t\ge0}$ which are independent from one parameter. This leads to a formula for $\mathbb E(\prod_{i=1}^N (y-X_{t,i}))$ in terms of classical Jacobi polynomials. For $t\to\infty$ this yields a corresponding formula for Jacobi ensembles and thus Aomoto's Selberg integral.
On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions
本文定义并计算了在复空间族上的beta积分族，当参数取特定值时，该积分族可给出Selberg积分的Dotsenko-Fateev复孪生形式。
Jacob's ladders, almost linear increments of the Hardy-Littlewood integral (1918) and their relations to the Selberg's formula (1946) and the Fermat-Wiles theorem
本文从哈代-利特尔伍德积分增量公式出发，建立了费马-怀尔斯定理的一个 ζ 等价形式，该形式仅用临界线上的黎曼 ζ 函数与雅可比梯表达。
$q$-Selberg extensions of Gasper's two stage $q$-beta integrals
本文通过 Aomoto 方法，将 Gasper 的两阶段 $q$-beta 积分公式推广到多维情形，并证明其退化为 Tarasov-Varchenko 公式。
In this article we consider the elliptic Selberg integral, which is a BC_n symmetric multivariate extension of the elliptic beta integral. We categorize the limits that are obtained as p->0, for given behavior of the parameters as p->0. This article is therefore the multivariate version of our earlier paper "Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions". The integrand of the elliptic Selberg integral is the measure for the BC_n symmetric biorthogonal functions introduced by the second author, so we also consider the limits of the associated bilinear form. We al…
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