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Register a new userOn generalized complete elliptic integrals and modular functions
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This paper deals with generalized elliptic integrals and generalized modular functions. Several new inequalities are given for these and related functions.
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The close form of some integrals involving recently developed generalized k-Struve functions is obtained. The outcome of these integrations is expressed in terms of generalized Wright functions. Several special cases are deduced which lead to some known results.
This is a brief overview of the status of the theory of elliptic hypergeometric functions to the end of 2012 written as a complementary chapter to the Russian edition of the book by G.E. Andrews, R. Askey, and R. Roy, Special Functions, Encycl. of Math. Appl. 71, Cambridge Univ. Press, 1999.
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 $pto 0$ degenerations.
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 Eulers and Selbergs 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.
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 authors habilitation thesis [Spi7] containing a more detailed account of the subject.
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