Well-poised Hypergeometric Transformations of Euler-type Multiple Integrals

Several new multiple-integral representations are proved for well-poised hypergeometric series and integrals. The results yield, in particular, transformations of the multiple integrals that cannot be achieved by evident changes of variable. All this generalizes some classical results of Whipple and Bailey in analysis and, on the other hand, certain analytic constructions with known connection to irrationality proofs for values of Riemann’s zeta function at positive integers. 1. Motivation from number theory In 1979, Apéry showed [2] that the values of ζ(s) = ∑∞ n=1 n −s at s = 2 and s = 3 are irrational. Although the irrationality of ζ(2) = π/6 was known, Apéry’s novel method for proving irrationality (‘accélération de la convergence’, cf. [8]) gave a new proof for ζ(2). In the same year, Beukers [7] suggested the Euler-type multiple integrals J2,n = ∫∫ [0,1]2 xn(1− x)nyn(1− y) (1− xy)n+1 dx dy (1)