A Whipple _7F_6 Formula Revisited

well-known formula of Whipple relates certain hypergeometric values _7F_6(1) and _4F_3(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 _7F_6(1) in Whipple’s formula to the periods of these modular forms.