Beta Integral Research Articles

Complex and rational hypergeometric functions on root systems

We consider some new limits for the elliptic hypergeometric integrals on root systems. After the degeneration of elliptic beta integrals of type I and type II for root systems An and Cn to the hyperbolic hypergeometric integrals, we apply the limit ω1→−ω2 for their quasiperiods (corresponding to b→i in the two-dimensional conformal field theory) and obtain complex beta integrals in the Mellin–Barnes representation admitting exact evaluation. Considering type I elliptic hypergeometric integrals of a higher order obeying nontrivial symmetry transformations, we derive their descendants to the level of complex hypergeometric functions and prove the Derkachov–Manashov conjectures for functions emerging in the theory of non-compact spin chains. We describe also symmetry transformations for a type II complex hypergeometric function on the Cn-root system related to the recently derived generalized complex Selberg integral. For some hyperbolic beta integrals we consider a special limit ω1→ω2 (or b→1) and obtain new hypergeometric identities for sums of integrals of rational functions.

Notes on $q$-Partial Differential Equations for $q$-Laguerre Polynomials and Little $q$-Jacobi Polynomials

This article defines two common $q$-orthogonal polynomials: homogeneous $q$-Laguerre polynomials and homogeneous little $q$-Jacobi polynomials. They can be viewed separately as solutions to two $q$-partial differential equations. Furthermore, an analytic function satisfies a certain system of $q$-partial differential equations if and only if it can be expanded in terms of homogeneous $q$-Laguerre polynomials or homogeneous little $q$-Jacobi polynomials. As applications, several generalized Ramanujan $q$-beta integrals and Andrews-Askey integrals are obtained.

Extended Exton’s Triple and Horn’s Double Hypergeometric Functions and Associated Bounding Inequalities

This paper introduces extensions H4,p and X8,p of Horn’s double hypergeometric function H4 and Exton’s triple hypergeometric function X8, taking into account recent extensions of Euler’s beta function, hypergeometric function, and confluent hypergeometric function. Among the numerous extended hypergeometric functions, the primary rationale for choosing H4 and X8 is their comparable extension type. Next, we present various integral representations of Euler and Laplace types, Mellin and inverse Mellin transforms, Laguerre polynomial representations, transformation formulae, and a recurrence relation for the extended functions. In particular, we provide a generating function for X8,p and several bounding inequalities for H4,p and X8,p. We explore the utilization of the H4,p function within a probability distribution. Most special functions, such as the generalized hypergeometric function, the Beta function, and the p-extended Beta integral, exhibit natural symmetry.

General closed-form expressions for the three-dimensional vibrations of elastic bodies using the Ritz method

A novel Ritz-β approach is proposed for accurately determining the vibrational frequencies of a body based on its three-dimensional elasticity. In the conventional Ritz method, when admissible 3D displacement fields are expressed as either simple algebraic, Chebyshev, or Legendre polynomials, the volumetric integrals are transformed into beta integrals through a change of variables and then furthered simplified to beta functions and gamma functions; the resulting stiffness and mass matrices are expressed in a closed-form and the computational time is significantly decreased. The proposed approach does not involve the intensive computational procedures employed in the classical finite element, finite difference, and spectral approximation theories. Furthermore, it does not involve the kinematic constraints employed in classical beam, plate, and shell theories. The correctness and accuracy of the proposed approach were confirmed through comprehensive convergence studies and comparisons with existing results.

On Some Properties of Leibniz's Triangle

One of the Greatest mathematicians of all time, Gotfried Leibniz, introduced amusing triangular array of numbers called Leibniz's Harmonic triangle similar to that of Pascal's triangle but with different properties. I had introduced entries of Leibniz's triangle through Beta Integrals. In this paper, I have proved that the Beta Integral assumption is exactly same as that of entries obtained through Pascal's triangle. The Beta Integral formulation leads us to establish several significant properties related to Leibniz's triangle in quite elegant way. I have shown that the sum of alternating terms in any row of Leibniz's triangle is either zero or a Harmonic number. A separate section is devoted in this paper to prove interesting results regarding centralized Leibniz's triangle numbers including obtaining a closed expression, the asymptotic behavior of successive centralized Leibniz's triangle numbers, connection between centralized Leibniz's triangle numbers and Catalan numbers as well as centralized binomial coefficients, convergence of series whose terms are centralized Leibniz's triangle numbers. All the results discussed in this section are new and proved for the first time. Finally, I have proved two exceedingly important theorems namely Infinite Hockey Stick theorem and Infinite Triangle Sum theorem. Though these two theorems were known in literature, the way of proving them using Beta Integral formulation is quite new and makes the proof short and elegant. Thus, by simple re-formulation of entries of Leibniz's triangle through Beta Integrals, I have proved existing as well as new theorems in much compact way. These ideas will throw a new light upon understanding the fabulous Leibniz's number triangle.

On the Askey–Wilson polynomials and a $q$-beta integral

On the Askey–Wilson polynomials and a $q$-beta integral

The Endless Beta Integrals

We consider a special degeneration limit $\omega_1\to - \omega_2$ (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 hyperbolic gamma function) is discovered in the limit $\omega_1\to \omega_2$ (or $b\to 1$).

The modular group and a hyperbolic beta integral

The modular group and a hyperbolic beta integral

Elliptic extension of Gustafson's q-integral of type G2

The evaluation formula for an elliptic beta integral of type G2 is proved. The integral is expressed by a product of Ruijsenaars' elliptic gamma functions, and the formula includes that of Gustafson's q-beta integral of type G2 as a special limiting case as p→0. The elliptic beta integral of type BC1 by van Diejen and Spiridonov is effectively used in the proof of the evaluation formula.

The modular group and a hyperbolic beta integral

The modular group and a hyperbolic beta integral

The rarefied elliptic Bailey lemma and the Yang–Baxter equation

An elliptic Bailey lemma is formulated on the basis of the univariate rarefied elliptic beta integral. It leads to a generalized operator star-triangle relation and a new solution of the Yang–Baxter equation written as an integral operator with a rarefied elliptic hypergeometric kernel.

Askey–Wilson polynomials and a double $q$-series transformation formula with twelve parameters

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 Ismail as a special case

From rarefied elliptic beta integral to parafermionic star-triangle relation

We consider the rarefied elliptic beta integral in various limiting forms. In particular, we obtain an integral identity for parafermionic hyperbolic gamma functions which describes the star-triangle relation for parafermionic Liouville theory.

Rarefied elliptic hypergeometric functions

Two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved. They generalize evaluations of the type I and II elliptic beta integrals attached to the root system Cn. In a special n=1 case, the simplest p→0 limit is shown to lead to a new class of q-hypergeometric identities. Symmetries of a rarefied elliptic analogue of the Euler–Gauss hypergeometric function are described and the respective generalization of the hypergeometric equation is constructed. Some extensions of the latter function to Cn and An root systems and corresponding symmetry transformations are considered. An application of the rarefied type II Cn elliptic hypergeometric function to some eigenvalue problems is briefly discussed.

Two Ramanujan’s formulas and normal distribution

In this paper, we point out the relationship between the two Ramanujan’s formulas for Beta integrals and normal distribution. Some of the applications are also given.

On the extended Appell-Lauricella hypergeometric functions and their applications

The main object of this paper is to present a systematic introduction to the theory and applications of the extended Appell-Lauricella hypergeometric functions defined by means of the extended beta function and extended Dirichlet?s beta integral. Their connections with the Laguerre polynomials, the ordinary Lauricella functions and the Srivastava-Daoust generalized Lauricella functions are established for some specific paramters. Furthermore, by applying the various methods and known formulas (such as fractional integral technique; some results of the Lagrange polynomials), we also derive some elegant generating functions for these new functions.

On Durrmeyer-type generalization of (p, q)-Bernstein operators

In this paper, we introduce (p, q)-Bernstein Durrmeyer operators. We define (p, q)-beta integral and use it to obtain the moments of the operators. We obtain uniform convergence of the operators by using Korovkin’s theorem. We estimate direct results of the operators by means of modulus of continuity and Peetre K-functional. Finally, we find Voronovskaya-type theorem for the operators.

Elliptic genera and characteristic q-series of superconformal field theory

We analyze the characteristic series, the KO series and the series associated with the Witten genus, and their analytic forms as the q-analogs of classical special functions (in particular q-analog of the beta integral and the gamma function). q-Series admit an analytic interpretation in terms of the spectral Ruelle functions, and their relations with appropriate elliptic modular forms can be described. We show that there is a deep correspondence between the characteristic series of the Witten genus and KO characteristic series, on one side, and the denominator identities and characters of N=2 superconformal algebras, and the affine Lie (super)algebras on the other. We represent the characteristic series in the form of double series using the Hecke–Rogers modular identity.

Evaluation of matrix-variate gamma and beta integrals as multiple integrals and Kober fractional integral operators in the complex matrix variate case

Explicit evaluations of matrix-variate gamma and beta integrals in the complex domain by using conventional procedures is extremely difficult. Such an evaluation will reveal the structure of these matrix-variate integrals. In this article, explicit evaluations of matrix-variate gamma and beta integrals in the complex domain for the order of the matrix p=1,2 are given. Then fractional integral operators of the Kober type are given for some specific cases of the arbitrary function. A formal definition of fractional integrals in the complex matrix-variate case was given by the author earlier as the M-convolution of products and ratios, where Kober operators become a special class of fractional integral operators.

Elliptic Hypergeometry of Supersymmetric Dualities II. Orthogonal Groups, Knots, and Vortices

We consider Seiberg electric-magnetic dualities for 4d $\mathcal{N}=1$ SYM theories with SO(N) gauge group. For all such known theories we construct superconformal indices (SCIs) in terms of elliptic hypergeometric integrals. Equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals. In particular, we describe a number of new elliptic beta integrals associated with the s-confining theories with the spinor matter fields. Reductions of some dualities from SP(2N) to SO(2N) or SO(2N+1) gauge groups are described. Interrelation of SCIs and the Witten anomaly is briefly discussed. Possible applications of the elliptic hypergeometric integrals to a two-parameter deformation of 2d conformal field theory and related matrix models are indicated. Connections of the reduced SCIs with the state integrals of the knot theory, generalized AGT duality for (3+3)d theories, and a 2d vortex partition function are described.