A Bailey Tree for Integrals

Series and integrals of hypergeometric type have many applications in mathematical physics. Investigation of the corresponding special functions can therefore be relevant for various computations in theoretical models of reality. Bailey chains provide powerful tools for generating infinite sequences of summation or transformation formulas for hypergeometric type series. For a review of the corresponding formalism and many q-series identities obtained with its help, see, e.g., [1], [2]. Recently, we extended the scope of applications of this technique to elliptic hypergeometric series [3]. More precisely, an elliptic generalization of the Andrews’ well-poised Bailey chain [1] has been found and its simplest consequences have been analyzed. This lifted also one of the Andrews-Berkovich identities [4] to the elliptic level. During the work on [3], we came to a principal conclusion that there must exist Bailey chains for integrals, but the first attempts to build a simple example of such a chain were not successful. In [5], a Bailey type symmetry transformation was constructed for a pair of elliptic hypergeometric integrals. This result gives the tools necessary for an appropriate generalization of the elliptic Bailey chain technique of [3]. In the present note, we construct two examples of Bailey chains for integrals directly at the level of the most complicated known type of beta integrals of one variable, namely, the elliptic beta integral of [6]. Being used together, they form a binary tree of identities for elliptic hypergeometric integrals. We denote by T the positively oriented unit circle and take two base variables q, p, |q|, |p| < 1, and five complex parameters tm,m = 0, . . . , 4, satisfying the inequalities |tm| < 1, |pq| < |A|, where A ≡ ∏4 r=0 tr. Then the elliptic beta integral