George Gasper, Michael Schlosser
We deduce several curious q-series expansions by applying inverse relations to certain identities for basic hypergeometric series. After rewriting some of these expansions in terms of q-integrals, we obtain, in the limit q -> 1, some curious beta-type integral evaluations which appear to be new.
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While investigating the properties of a galaxy model used in Stellar Dynamics, a curious integral identity was discovered. For a special value of a parameter, the identity reduces to a definite integral with a very simple symbolic value; but, quite surprisingly, all the consulted tables of integrals, and computer algebra systems, do not seem aware of this result. Here I show that this result is a special case ($n=0$ and $z=1$) of the following identity (established by elementary methods): $$ I_n(z)\equiv\int_0^1{{\rm K}(k) k\over (z+k^2)^{n+3/2}}dk = {(-2)^n\over (2n+1)!!} {d^n\over dz^n} {{\r…
By using the theory of the elliptic integrals a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect to the elliptic modulus and integral representations of several of the series in terms of the inverse Mellin transforms related to the Riemann zeta function. The relation with the corresponding case of the Voronoi summation formula is exhibited. The involved series are expressed in closed form in terms of complete elliptic integrals of the first and second ki…
We prove a novel type of inversion formula for elliptic hypergeometric integrals associated to a pair of root systems. Using the (A,C) inversion formula to invert one of the known C-type elliptic beta integrals, we obtain a new elliptic beta integral for the root system of type A. Validity of this integral is established by a different method as well.
Evaluation of beta integrals of Ramanujan type and integral representations for bilateral hypergeometric series
本文评估Ramanujan型伽马函数乘积积分，表示为双边超几何级数，并在可求和时化为beta积分，同时给出双边超几何级数的积分表示。
We present the evaluation of definite integrals in the classical table by I. S. Gradshteyn and I. M. Ryzhik that can be reduced to the beta function.
The table of Gradshteyn and Rhyzik contains some trigonometric integrals that can be expressed in terms of the beta function. We describe the evaluation of some of them.
We give elementary proofs of the univariate elliptic beta integral with bases $|q|, |p|<1$ and its multiparameter generalizations to integrals on the $A_n$ and $C_n$ root systems. We prove also some new unit circle multiple elliptic beta integrals, which are well defined for $|q|=1$, and their $p\to 0$ degenerations.
We present some elliptic beta integrals with a base parameter on the unit circle, together with their basic degenerations.
This research note deals with the evaluation of some generalized beta-type integral operators involving the multi-index Mittag-Leffler function $E_{ε_{i}),(ω_{i})}(z)$. Further, we derive a new family of beta-type integrals involving the product of a multi-index Mittag-Leffler function and a generating function of two variables. Some concluding remarks regarding our present investigation are briefly discussed in the last section.
The table of Gradshteyn and Ryzhik contains some integrals that can be expressed in terms of the incomplete beta function. We describe some elementary properties of this function and use them to check some of the formulas in the mentioned table.
A Tribute to Richard Askey, and an Expansive View of Some of his Favorite Beta Integrals
本文致敬 Richard Askey，并重新审视其钟爱的 Beta 积分，提出 Cauchy beta 积分在实对称矩阵空间上的多维推广，进而推导出 Cauchy--Selberg 积分的特例。
We construct 3 finite systems of $4-F-3$ hypergeometric orthogonal polynomials. The weights are 1) the weight defined by the $5-H-5$ Dougall summation formula; 2) the integrand in the Askey beta-integral; 3) the weight $w(s)=|p(s)/q(s)|^2$, where $p(s)=Γ(a+is)Γ(b+is)Γ(c+is)$ and $q(s)=Γ(d+is)Γ(2is)$. We also evaluate the integral of the function $w(s)$ using the Jacobi transform (the Olevsky transform); this integral also can be reduced to the Nassrallah--Rahman integral.
In this article we prove a new elliptic hypergeometric integral identity. It previously appeared (as a conjecture) in articles by Rains, and Spiridonov and Vartanov. Moreover it gives a different proof of an identity in another article by Rains. We also give some basic hypergeometric and classical limits of this identity. The classical limit gives identities (some known, some new) between generalizations of the Selberg integral.
A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and parameters is given. The constant factor related to the hypergeometric function and the beta function is proved to be the best possible. As applications, equivalent forms, the reverses, some particular examples, two kinds of Hardy-type inequalities, and operator expressions are considered.
The beta integral method proved itself as a simple nonetheless powerful method of generating hypergeometric identities at a fixed argument. In this paper we propose a generalization by substituting the beta density with a particular type of Meijer's G function. By application of our method to known transformation formulas we derive about forty hypergeometric identities, majority of which are believed to be new. We further apply some of these transformations to obtain several new summation formulas.
Accelerating the Hypergeometric Function with the Beta Integral to Derive New Infinite Series for $π$ and Values of the Gamma Function
本文通过beta积分加速超几何函数$2 F 1$，推导出$π$和Gamma函数值的新无穷级数，并基于初等反三角函数得出每项增加2.5位数字的紧凑类Ramanujan公式。
We give a direct evaluation of a curious integral identity, which follows from the work of Ismail and Valent on the Nevanlinna parametrization of solutions to a certain indeterminate moment problem.
Hyperbolic beta integrals are analogues of Euler's beta integral in which the role of Euler's gamma function is taken over by Ruijsenaars' hyperbolic gamma function. They may be viewed as $(q,\widetilde{q})$-bibasic analogues of the beta integral in which the two bases $q$ and $\widetilde{q}$ are interrelated by modular inversion, and they entail $q$-analogues of the beta integral for $|q|=1$. The integrals under consideration are the hyperbolic analogues of the Ramanujan integral, the Askey-Wilson integral and the Nassrallah-Rahman integral. We show that the hyperbolic Nassrallah-Rahman integ…
In this article we continue the work from arXiv:0902.0621. In that article Eric Rains and the present author considered the limits of the elliptic beta integral as p->0 while the parameters t_r have a p-dependence of the form t_r=u_rp^{α_r} (for fixed u_r and certain real numbers α_r). In this article we again consider such limits, but now we let p->0 along a geometric sequence p=xq^{sk} (for some integer s, while k -> \infty), and only allow α_r\in 2/s Z. These choices allow us to take many more limits. In particular we now also obtain bilateral basic hypergeometric series as possible limits,…
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…
First examples of matrix beta-integrals were discovered on 1930-50s by Siegel and Hua, in 60s Gindikin obtained multi-parametric series of such integrals. We discuss beta-integrals related to symmetric spaces, their interpolation with respect to the dimension of a ground field, and adelic analogs; also we discuss beta-integrals related to flag spaces.
We construct a family of measures on flag spaces (or, equivalently, on the spaces of upper-triangular matrices) compatible with respect to natural projections. We obtain an $n(n-1)/2$-parametric family of beta-integrals over space of upper-triangular matrices of size $n$.
We interpolate matrix beta-integrals of Siegel, Hua Loo Keng and Gindikin types with respect to dimension of the field. The domain of integration (Rayleigh triangles) imitates collections of all the eigenvalues of all the principal minors of a self-adjoint matrix. We also interpolate the Hua--Pickrell measures on inverse limits of symmetric spaces. Our family of integrals also contains the Selberg integral.
We use generating functions to express orthogonality relations in the form of $q$-beta integrals. The integrand of such a $q$-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal
We derive new reduction formulas for the incomplete beta function and the Lerch transcendent in terms of elementary functions. As an application, we calculate some new integrals. Also, we use these reduction formulas to test the performance of the algorithms devoted to the numerical evaluation of the incomplete beta function.
We define a general class of (multiple) integrals of hypergeometric type associated with the Jacobi theta functions. These integrals are related to theta hypergeometric series through the residue calculus. In the one variable case, we get theta function extensions of the Meijer function. A number of multiple generalizations of the elliptic beta integral [S2] associated with the root systems $A_n$ and $C_n$ is described. Some of the $C_n$-examples were proposed earlier by van Diejen and the author, but other integrals are new. An example of the biorthogonality relations associated with the elli…
We study the probability that one beta-distributed random variable exceeds the maximum of two others, allowing all three to have general parameters. This amounts to studying Euler transforms of products of two incomplete beta functions. We provide a closed form for the general problem in terms of Kampé de Fériet functions and a variety of simpler closed forms in special cases. The results are applied to derive the moments of the maximum of two independent beta-distributed random variables and to find inner products of incomplete beta functions. Restricted to positive integer parameters, our re…
The aim of this research paper is to demonstrate how one can obtain eleven new and interesting hypergeometric identities (in the form of a single result) from the old ones by mainly applying the well known beta integral method which was used successfully and systematically by Krattenthaler and Rao in their well known, very interesting research papers. The results are derived with the help of generalization of a quadratic transformation formula due to Kummer very recently obtained by Kim, et al. . Several identities including one obtained earlier by Krattenthaler and Rao follow special cases of…
In the present paper, we consider (p,q)-analogue of the Beta operators and using it, we propose the integral modification of the generalized Bernstein polynomials. We estimate some direct results on local and global approximation. Also, we illustrate some graphs for the convergence of (p,q)-Bernstein-Durrmeyer operators for different values of the parameters p and q using Mathematica package.
General modular quantum dilogarithm and beta integrals
本文研究了一般模量子双对数函数组成的单变量beta积分，证明了其精确求值公式，并指出该公式可表示特定3d超对称场论在一般压缩透镜空间上的配分函数，同时探讨了其在2d共形场论中的应用。
The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integral representation of the classical $_2F_1$ function. Based on a $q$-series transformation formula and the Nassrallah--Rahman integral we prove a $q$--beta integral which has twelve parameters, with several other results, both classical and new, included as special cases. This $q$-beta integral also allows us to derive a curious double $q$--series transformation formula, which includes one formula of Al--Salam and Ism…
There exist several multivariate extensions of the classical Sonine integral representation for Bessel functions of some index $μ+ ν$ with respect to such functions of lower index $μ.$ For Bessel functions on matrix cones, Sonine formulas involve beta densities $β_{μ,ν}$ on the cone and trace already back to Herz. The Sonine representations known so far on symmetric cones are restricted to continuous ranges $\Reμ, \Re ν> μ_0$, where the involved Beta densities are probability measures and the limiting index $μ_0\geq 0$ depends on the rank of the cone. It is zero only in the one-dimensional cas…
Bilateral discrete and continuous orthogonality relations in the $q^{-1}$-symmetric Askey scheme
本文在$q^{-1}$-对称Askey框架下计算了五类正交多项式的双边离散与连续正交关系，并由$q^{-1}$-Askey--Wilson多项式的连续正交关系导出$q$-beta积分，其在$q\ o 1^{-}$极限下对应一个Ramanujan型beta积分并给出两种证明。
$q$-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。
We consider a special degeneration limit $ω_1\to - ω_2$ (or $b\to {\rm i}$ in the context of $2d$ Liouville quantum field theory) for the most general univariate hyperbolic beta integral. This limit is also applied to the most general hyperbolic analogue of the Euler-Gauss hypergeometric function and its $W(E_7)$ group of symmetry transformations. Resulting functions are identified as hypergeometric functions over the field of complex numbers related to the ${\rm SL}(2,\mathbb{C})$ group. A new similar nontrivial hypergeometric degeneration of the Faddeev modular quantum dilogarithm (or hyperb…
We, by making use of elementary arguments, deduce integral representations of the Legendre chi function $χ_{s}(x)$ valid for $|z|<1$ and $\Re(s)>1$. Our earlier established results on the integral representations for the Riemann zeta function $ζ(2 n+1)$ and the Dirichlet beta function $β(2 n)$ ,$ n\in\mathbb{N}$, are direct consequence of these representations.
In this article we give a new transformation between elliptic hypergeometric beta integrals, which gives rise to a Weyl group symmetry of type F_4. The transformation is a generalization of a series transformation discovered by Langer, Schlosser, and Warnaar. Moreover we consider various limits of this transformation to basic hypergeometric functions obtained by letting p tend to 0.
Sums involving the digamma function connected to the incomplete beta function and the Bessel functions
本文闭式计算了若干与不完全 beta 函数或 Bessel 函数相关的 digamma 无穷和，并导出超几何函数约化、参数微分及新定积分等结果。
Explicit Formulas For Generalized Polylogarithmic Integrals, Euler Sums, And BBP-Type Series
本文研究含五参数的多对数积分闭式表达式，用Lerch超越函数统一表示，并讨论其与欧拉和及BBP型级数的联系。
Elliptic hypergeometric functions: integrals versus series
本文研究椭圆超几何函数，将椭圆贝塔积分表示为无穷级数组合，并讨论其收敛性及终止级数的渐近行为。
These lecture notes were written for a mini-course that was designed to introduce students and researchers to {\it $q$-series,} which are also called {\it basic hypergeometric series} because of the parameter $q$ that is used as a base in series that are ``{\it over, above or beyond}'' the {\it geometric series}. We start by considering $q$-extensions (also called $q$-analogues) of the binomial theorem, the exponential and gamma functions, and of the beta function and beta integral, and then progress on to the derivations of rather general summation, transformation, and expansion formulas, int…
Determining the Indeterminate: On the Evaluation of Integrals that connect Riemann's, Hurwitz' and Dirichlet's Zeta, Eta and Beta functions
本文研究了连接黎曼、Hurwitz和狄利克雷ζ、η和β函数的积分的评估，揭示了与本质奇点相关的不定性，并提出了新的符号和方法来处理这些积分。
We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transfo…
We derive a beta-integral over $\mathbb{Z}\times \mathbb{R}$ , which is a counterpart of the Dougall $_5H_5$-formula and of the de Branges--Wilson integral, our integral includes $_{10}H_{10}$-summation. For a derivation we use a two-dimensional integral transform related to representations of the Lorentz group, this transform is a counterpart of the Olevskii index transform (a synonym: Jacobi transform).
We obtain two new Thomae-type transformations for hypergeometric series with r pairs of numeratorial and denominatorial parameters differing by positive integers. This is achieved by application of the so-called Beta integral method developed by Krattenthaler and Rao [Symposium on Symmetries in Science (ed. B. Gruber), Kluwer (2004)] to two recently obtained Euler-type transformations. Some special cases are given.
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