# Hypergeometric Functions over Finite Fields

## Jenny Fuselier

### 2019

We focus on the finite field analogues of classical transformation and evaluation formulas based on the papers mentioned above, especially by Greene and by McCarthy

## Highlights

• We focus on the finite field analogues of classical transformation and evaluation formulas based on the papers mentioned above, especially by Greene and by McCarthy
• We describe the foundation for the finite field analogue of the hypergeometric function
• There are many evaluation and transformation formulas for the classical 2F1 hypergeometric functions that are incredibly useful for obtaining higher transformation formulas, computing periods, or proving supercongruences
• This technique is employed in work of the second, third, and fourth authors, among others. These evaluation formulas are obtained in various ways; for example, see
• In this subsection we address the relations between the period 2P1 hypergeometric functions corresponding to independent solutions of the hypergeometric differential equation, and Kummer’s 24 relations
• We have used our methods to prove finite field analogues of numerous classical formulas including 9 quadratic or higher transformation formulas, 11 evaluation formulas, and 3 algebraic identities, among other formulas