Continuous biorthogonality of an elliptic hypergeometric function

Abstract

Abstract. A family of continuous biorthogonal functions related to an elliptic analog of the Gauss hypergeometric function is constructed. The key tools used for that are the elliptic beta integral and the integral Bailey chain introduced earlier by the author. The relationship with the Sklyanin algebra and elliptic analogs of the Faddeev modular double are discussed in detail. §1. Introduction Classical and quantum completely integrable systems serve as a rich source of special functions. The complexity of these systems correlates with the complexity of solutions of the Yang-Baxter equation -rational, trigonometric, or elliptic [22]. In the theory of special functions this structural hierarchy is reflected in the existence of the plain and q-hypergeometric functions [1] and their elliptic generalizations [19]. One of our aims in the present paper is to clarify certain questions about this correspondence at the elliptic level.In [17,19], the author introduced an elliptic analog of the Gauss hypergeometric function V (t 1 , . . . , t 8 ) that generalizes many special functions of hypergeometric type obeying "classical" properties [1]. In particular, it was shown that this function exhibits symmetry transformations tied to the exceptional root system E 7 and satisfies the elliptic hypergeometric equation. We start this work with showing in §2 that the V -function satisfies a simple biorthogonality relation corresponding to the continuous values of a spectral parameter. For that we use the elliptic beta integral [16] and an integral analog of the Bailey chains introduced in [18]. Our construction can be viewed as an integral generalization of Rosengren's approach to the elliptic 6j-symbols [11].In §3, we investigate the generalized eigenvalue problem for a second order finite difference operator D (a, b, c, d; p; q). The relationship of such problems with biorthogonality relations was considered in [23]. A problem similar to ours was investigated earlier in [10,11] for the discrete spectrum. Its general solution determines a basis in some space of meromorphic functions. For a simple scalar product, we derive the dual basis defined by solutions of a similar spectral problem for conjugate operators. As a result, the overlap of the basis vectors with their duals coincides with the V -function. In §4, we derive a new contiguous relation for the elliptic hypergeometric function and the biorthogonality condition associated with that.It turns out that the V -function is directly related to the Sklyanin algebra, one of the central objects in the quantum inverse scattering method [14,15]. This relationship is established in §5 on the basis of the observation, due to Rains [9,12], that the D-operator can be represented as a linear combination of all four generators of the Sklyanin algebra.2000 Mathematics Subject Classification. Primary 33C75, 81R12.

Essays on the theory of elliptic hypergeometric functions

“…On the other hand, the classical equations of motion for this Hamiltonian with the modular parameter τ considered as a time variable leads to the Painlevé VI equation [83]. The Doperator itself appears to be related to the van Diejen model [62,79]. All this and the connection with elliptic hypergeometric functions described below demonstrate some mathematical universality of the operator (8.1).…”

“…In paper [79], it was shown that the V -function with general set of continuous parameters obeys also some biorthogonality relations characteristic to the continuous spectra.…”

“…The standard eigenvalue problem Dψ = λψ appears to be very complicated, since it represents a difference analogue of the Heun equation [79]. On the one hand, it is known that the eigenvalue problem for the one particle Hamiltonian of the Inozemtsev model [65] is equivalent to the Heun equation.…”

“…for |q| < 1 we obtain [79] f (z; w; a, b; For z → z −1 invariant functions, ψ(z) = ψ(z −1 ), we define the scalar product…”

“…The latter double can be obtained from the second case in the limit Im(τ ) → +∞ (in the first case this limit is not defined) [79].…”