On some New Modular Equations and their Applications to Continued Fractions

In this paper, we obtain some new modular equations of degree2. We obtain several general formulas for the explicit evaluations of the Ramanujan's theta{function. As an application, we establish somenew modular relations for Ramanujan{Gollnitz{Gordon continued frac-tion H(q) with H(qn), Ramanujan{Selberg continued fraction V (q) with V (qn) and Eisenstein continued fraction E(q) with E(qn) for n =6; 10; 14 and 16. We also establish their explicit evaluations

On some new modular equations of degree 9 and their applications

In this paper, we establish several new modular equations of degree 9 using Ramanujan's modular equations. We also establish several new general formulas to compute the values for r 9,n and râ² 9. As an application, we establish explicit evaluations of Ramanujan's remarkable product of theta-functions

Some New Modular Equations of Degree 2 Akin to Ramanujan

In this paper, we obtain some new modular equations of degree 2 for the ratios of Ramanujan's theta-function f and also establish the general formulas for their explicit evaluations. As an application, we establish some new modular relations for Ramanujan-Göllnitz-Gordon continued fraction H(q) with H(qn/2), Ramanujan-Selberg continued fraction V(q) with V(qn/2) and Eisenstein continued fraction E(q) with E(qn/2) for n=3, 5 and 7

Aspects of elliptic hypergeometric functions

General elliptic hypergeometric functions are defined by elliptic hypergeometric integrals. They comprise the elliptic beta integral, elliptic analogues of the Euler-Gauss hypergeometric function and Selberg integral, as well as elliptic extensions of many other plain hypergeometric and $q$-hypergeometric constructions. In particular, the Bailey chain technique, used for proving Rogers-Ramanujan type identities, has been generalized to integrals. At the elliptic level it yields a solution of the Yang-Baxter equation as an integral operator with an elliptic hypergeometric kernel. We give a brief survey of the developments in this field.Comment: 15 pp., 1 fig., accepted in Proc. of the Conference "The Legacy of Srinivasa Ramanujan" (Delhi, India, December 2012

Notes on the Riemann Hypothesis

These notes were written from a series of lectures given in March 2010 at the Universidad Complutense of Madrid and then in Barcelona for the centennial anniversary of the Spanish Mathematical Society (RSME). Our aim is to give an introduction to the Riemann Hypothesis and a panoramic view of the world of zeta and L-functions. We first review Riemann's foundational article and discuss the mathematical background of the time and his possible motivations for making his famous conjecture. We discuss some of the most relevant developments after Riemann that have contributed to a better understanding of the conjecture.Comment: 2 sections added, 55 pages, 6 figure

Classical elliptic hypergeometric functions and their applications

General theory of elliptic hypergeometric series and integrals is outlined. Main attention is paid to the examples obeying properties of the "classical" special functions. In particular, an elliptic analogue of the Gauss hypergeometric function and some of its properties are described. Present review is based on author's habilitation thesis [Spi7] containing a more detailed account of the subject.Comment: 42 pages, typos removed, references update