Geodesic
Determinants of Elliptic Hypergeometric Integrals
Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 4, pp. 67-86.

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We start from an interpretation of the $BC_2$-symmetric “Type I” (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation and then generalize this construction to higher-dimensional integrals and higher-order hypergeometric functions. This allows us to prove the corresponding formulas for the elliptic beta integral and symmetry transformation in a new way, by proving that both sides satisfy the same difference equations and that these difference equations satisfy a needed Galois-theoretic condition ensuring the uniqueness of the simultaneous solution.
Keywords: elliptic hypergeometric function, difference equation, determinant, difference Galois theory. 
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E. M. Rains; V. P. Spiridonov. Determinants of Elliptic Hypergeometric Integrals. Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 4, pp. 67-86. http://geodesic.mathdoc.fr/item/FAA_2009_43_4_a5/

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