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Essays on the theory of elliptic hypergeometric functions. (English. Russian original) Zbl 1173.33017

Russ. Math. Surv. 63, No. 3, 405-472 (2008); translation from Usp. Mat. Nauk 63, No. 3, 3-72 (2008).
The present paper is a survey of the main results of the theory of elliptic hypergeometric functions. These functions form a class of special functions of mathematical physics. Proofs are given for the most general known univariate exact integration formula generalizing Euler’s beta integral. In Theorem 1 the elliptic beta integral, discovered by the author in [V. P. Spiridonov, Russ. Math. Surv. 56, No. 1, 185–186 (2001); translation from Usp. Mat. Nauk 56, No.1, 181-182 (2001; Zbl 0997.33009)], is described. Let \(t_j\), \(j=1,\dotsc,6\) be six complex parameters and let be \(p\) and \(q\) two basic variables satisfying the conditions \(|p|\), \(|q|\), \(|t_j|<1\) and \(\prod_{j=1}^6t_j=pq\) (the balancing condition). Then the following equality is valid:
\[ \kappa\int_{\mathbb T}\frac{\prod_{j=1}^6\Gamma(t_jz^{\pm1};p,q)} {\Gamma(z^{\pm2};p,q)}\frac{dz}{z}=\prod_{1\leq j<k \leq6} \Gamma(t_jt_k; p,q), \]
where \(\Gamma(z;p,q)=(pqz^{-1};p,q)_\infty/(z;p,q)_\infty\), \((z;p,q)_\infty=\prod_{j,k=0}^\infty(1-zp^jq^k)\), \(\kappa= (p,p)_\infty (q,q)_\infty/(i4\pi)\) and \(\mathbb T\) denotes the unit circle with positive orientation. An elliptic analogue of the Gauss hypergeometric function.
\[ V(t_1,\dotsc,t_8;p,q)=\kappa\int_{\mathbb T}\frac{\prod_{j=1}^8 \Gamma(t_jz^{\pm1};p,q)} {\Gamma(z^{\pm2};p,q)}\frac{dz}{z}, \]
is given together with the elliptic hypergeometric equation for it, where the balancing condition has the form \(\prod_{j=1}^8t_j= p^2q^2\). A biorthogonality relation for this function and its particular subcases are described. The known elliptic integrals on root systems are listed and symmetry transformations are considered for the corresponding hight-order elliptic hypergeometric functions.

MSC:

33D67 Basic hypergeometric functions associated with root systems
33E05 Elliptic functions and integrals
33D70 Other basic hypergeometric functions and integrals in several variables
33C75 Elliptic integrals as hypergeometric functions
33B15 Gamma, beta and polygamma functions

Citations:

Zbl 0997.33009
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