Elliptic Hypergeometric Functions

This is a brief survey of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. A proof is given of the most general known univariate exact integration formula generalizing Euler's beta integral. It 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. The known elliptic beta integrals on root systems are listed, and symmetry transformations are considered for the corresponding higher-order elliptic hypergeometric functions.

Rarefied elliptic hypergeometric functions

We give a survey of elliptic hypergeometric functions associated with root systems, comprised of three main parts. The first two form in essence an annotated table of the main evaluation and transformation formulas for elliptic hypergeometric integeral and series on root systems. The third and final part gives an introduction to Rains' elliptic Macdonald-Koornwinder theory (in part also developed by Coskun and Gustafson).

We give a brief survey on the three-level hierarchy of hypergeometric series: ordinary (or rational) hypergeometric series, basic (or trigonometric, or q-) hypergeometric series, and, on the top level, elliptic hypergeometric series. The three different types of hypergeometric series satisfy a number of well known identities.Arguably the most fundamental identity for single series is the 10-V-9 summation formula (discovered by Frenkel and Turaev in 1997). This powerful elliptic hypergeometric series identity includes many of the other classical summations as special or limiting cases, including the well-known q-Chu-Vandermonde summation, an identity which can be easily interpreted combinatorially and corresponds to a convolution of q-binomial coefficients. We give a similar combinatorial interpretation for the 10-V-9 summation, now as a convolution of elliptic binomial coefficients. This is achieved by a weighted enumeration of lattice paths in a suitable lattice path model where we choose the weights to be specific elliptic functions. The weighted lattice path interpretation corresponds to a formulation entirely in algebraic terms; we describe an algebra of so called "elliptic commuting variables" in which the elliptic binomial coefficients appear as the normal form coefficients of a binomial. While our focus in this talk is on identities for basic and elliptic hypergeometric series (and part of our motivation for giving this talk is to make the area of elliptic hypergeometric series better known, in particular to researchers working in algebraic combinatorics), other suitable choices of the weights in our convolutions of weighted binomial coefficients yield convolutions of symmetric functions.

Proof that the Natural Logarithm Can Be Represented by the Gaussian Hypergeometric Function

Limits of elliptic hypergeometric biorthogonal functions

We consider an elliptic analogue of the Gauss hypergeometric function and two of its multivariate generalizations. We describe their relation to elliptic beta integrals, the exceptional Weyl group E7, the elliptic hypergeometric equation, and Calogero-Sutherland-type models.

Lecture notes for a course given at the summer school OPSF-S6, College Park, Maryland, 11-15 July 2016. In these lecture notes I give an elementary introduction to elliptic hypergeometric functions. I focus on motivating the main ideas and constructions, rather than giving a comprehensive survey. The lectures include a brief explanation of the historical origin of elliptic hypergeometric functions in the context of solvable lattice models. In particular, I give a new proof of the fact that fused Boltzmann weights for the elliptic solid-on-solid model can be expressed as elliptic hypergeometric sums.

5 - Indefinite Integrals of Special Functions

We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental.

In dieser Arbeit behandeln wir exakte, nicht-perturbative Ergebnisse, die mithilfe der superkonformen Index-Technik, in supersymmetrischen Eichtheorien mit vier Superladungen (d. h. N=1 Supersymmetrie in vier Dimensionen und N=2 in drei Dimensionen) gewonnen wurden. Wir benutzen die superkonforme Index-Technik um mehrere Dualitäts Vermutungen in supersymmetrischen Eichtheorien zu testen. Wir führen Tests der dreidimensionalen Spiegelsymmetrie und Seiberg ähnlicher Dualitäten durch. Das Ziel dieser Promotionsarbeit ist es moderne Fortschritte in nicht-perturbativen supersymmetrischen Eichtheorien und ihre Beziehung zu mathematischer Physik darzustellen. Im Speziellen diskutieren wir einige interessante Identitäten der Integrale, denen einfache und hypergeometrische Funktionen genügen und ihren Bezug zu supersymmetrischen Dualitäten in drei und vier Dimensionen. Methoden der exakten Berechnungen in supersymmertischen Eichtheorien sind auch auf integrierbare statistische Modelle anwendbar. Dies wird im letzten Kapitel der vorliegenden Arbeit behandelt.

In the genus one case, we make explicit some constructions of Veech on flat surfaces and generalize some geometric results of Thurston about moduli spaces of flat spheres as well as some equivalent ones but of an analytico-cohomological nature of Deligne-Mostow, which concern the monodromy of Appell-Lauricella hypergeometric functions. In the twin paper arXiv:1604.01812, we follow Thurston's approach and study moduli spaces of flat tori with conical singularities and prescribed holonomy by means of geometrical methods relying on surgeries for flat surfaces. In the present paper, we study the same objects making use of analytical and cohomological methods, more in the spirit of Deligne-Mostow's paper.

We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hypergeometric series. Single and multivariable elliptic hypergeometric series are considered in detail. A characterization theorem for a single variable totally elliptic hypergeometric series is proved.

The Indian govt. celebrated 125𝑡ℎ anniversary of the great Mathematician of Indian soil Srinivasa Ramanujan on 22 December in the year 2012. Without any formal education and extreme poverty conditions, he emerged as one of great mathematician of India. His mathematical ideas transformed and reshaped 20th century mathematics and their ideas are inspiration for 21𝑠𝑡 century mathematicians. Srinivasa Ramanujan made substantial contributions to the analytical theory of numbers and worked on ‘elliptic functions’, ‘continued fractions’, and ‘infinite series’. He was a great Mathematician, who became world famous at the tender age of twenty-six. He was born into a family that had a humble background and that had no distinguished professional achievement, yet his mathematical ideas transformed and reshaped century mathematics and continues to inspire modern day mathematicians. Considered to be a mathematical genius, Srinivasa Ramanujan, was regarded at par with the likes of Leonhard Euler and Carl Jacobi. In spite of having almost no formal training in mathematics, Ramanujan’s knowledge of mathematics was astonishing. Even though he had no knowledge of the modern developments in the subject, he effortlessly worked out the Riemann series, the elliptic integrals, hypergeometric series, and the functional equations of the zeta function. The purpose of this paper is to introduce some of the contributions of Srinivasa Ramanujan in the field of mathematics.

We give a brief account of the key properties of elliptic hypergeometric integrals—a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic generalization of Euler’s and Selberg’s beta integrals, elliptic analogue of the Euler–Gauss hypergeometric function and some multivariable elliptic hypergeometric functions on root systems. The elliptic Fourier transformation and corresponding integral Bailey lemma technique is outlined together with a connection to the star-triangle relation and Coxeter relations for a permutation group. We review also the interpretation of elliptic hypergeometric integrals as superconformal indices of four dimensional supersymmetric quantum field theories and corresponding applications to Seiberg type dualities.