Transformations of elliptic hypergeometric integrals. (English) Zbl 1209.33014

The elliptic hypergeometric integrals of different dimensions associated with the root systems \(A_n\) and \(BC_n\) are considered.
A pair of integral transformations \(BC_n \to BC_m\) and \(A_n \to A_m\) is proved. In the papers by J. F. van Diejen and V. P. Spiridonov [Math. Res. Lett. 7, No. 5–6, 729–746 (2000; Zbl 0981.33013); Int. Math. Res. Not. 2001, No. 20, 1083–1110 (2001; Zbl 1010.33010)] and V. P. Spiridonov [St. Petersbg. Math. J. 15, No. 6, 929–967 (2004) and Algebra Anal. 15, No. 6, 161–215 (2003; Zbl 1071.33011)] two types of integrals “Type I” and “Type II” are conjectured. The integral of “Type I” can be considered as the generalization of the integral identity of G. W. Anderson in [Forum Math. 3, No. 4, 415–417 (1991; Zbl 0723.33002)] connected with the Selberg integral which the “Type II” integral generalizes. A random matrix interpretation of the Anderson integral permits to generalize it to the elliptic level. The method used in the proof of the integral identity permits to obtain a relation between the \(n\)-dimensional integral with several parameters to an \(m\)-dimensional integral with transformed parameters. When \(m=0\) this gives the van Diejen-Spiridonov integral.
Using the fact that the “Type II” integral follows as a corollary of the “Type I” integral, the author constructs a family of functions satisfying a biorthogonality relation with respect to the “Type II” integral. This leads to generalizations of a series of results of Koornwinder, Askey-Wilson, Spiridonov and Zhedanov, Rahman. The author’s discussion of the biorthogonal functions begins by describing three kinds of difference operators and the spaces of functions on which they act. Then the corresponding integral operators are constructed. Combining these ingredients, the author constructs the biorthogonal functions and describes their main properties. The biorthogonal functions are constructed as the images of suitable sequences of difference and integral operators. The biorthogonality is defined with respect to a generalization of the Koornwinder density which bases on Okunkov’s \(BC_n\)-symmetric interpolation polynomials. Unlike in the Koornwinder case, at the elliptic level the required interpolation functions are special cases of the biorthogonal functions. These interpolation functions satisfy a number of generalized hypergeometric identities of Warnaar type, more exactly, they satisfy multivariate analogues of Jackson’s summation and Bailey transformations.
As an example, the eight-parameter version of the integral of “Type II” is considered, which is invariant with respect to an action of the Weyl group \(E_7\). In an appendix the singularities of an integral of a meromorphic function are investigated. These results are applied to obtain informations about the singularities of the integrals considered in the paper.


33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
33D67 Basic hypergeometric functions associated with root systems
33E05 Elliptic functions and integrals
33D70 Other basic hypergeometric functions and integrals in several variables
33C70 Other hypergeometric functions and integrals in several variables
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