  We examine hypergeometric functions in the finite field, p-adic and classical
settings. In each setting, we prove a formula which splits the hypergeometric
function into a sum of lower order functions whose arguments differ by roots of
unity. We provide multiple applications of these results, including new
reduction and summation formulas for finite field hypergeometric functions,
along with classical analogues; evaluations of special values of these
functions which apply in both the finite field and p-adic settings; and new
relations to Fourier coefficients of modular forms.
