A Whipple 7F6 Formula Revisited

A well-known formula of Whipple relates certain hypergeometric values ₇F₆(1) and ₄F₃(1). In this paper, we revisit this relation from the viewpoint of the underlying hypergeometric data HD, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple’s formula when the hypergeometric data HD are primitive and self-dual. If the data are also defined over Q, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of ℓ-adic representations of the absolute Galois group of Q attached to HD. For specialized choices of HD, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values ₇F₆(1) in Whipple’s formula to the periods of these modular forms.