In the literature, finite field hypergeometric functions were originally defined by Greene as analogues of classical hypergeometric series and were also introduced by Katz about the same time. Greene’s work includes an extensive catalogue of transformation and summation formulas for these functions, mirroring those in the classical case. These functions also have a nice character sum representation, and so the transformation and summation formulas can be interpreted as relations to simplify and evaluate complex character sums. Using character sums to count the number of solutions to equations over finite fields is a long established practice, and finite field hypergeometric functions also naturally lend themselves to this endeavor. Following the modularity theorem, and, by then, known links between finite field hypergeometric functions and elliptic curves, many authors began examining links between these functions and Fourier coefficients of modular forms. More recently, finite field hypergeometric functions have played a central role in the theory of hypergeometric motives, which has led to increased interest in the functions and their properties.
The main purpose of this paper is to generalize an earlier theorem, namely Theorem 1.2 stated in the paper. In the presented paper, the authors examined hypergeometric functions in the finite field, \(p\)-adic and classical settings. In each setting, the authors proved a formula which splits the hypergeometric function into a sum of lower order functions whose arguments differ by roots of unity. They acquired 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.