1. The Gauss Function 2. The Generalized Gauss Function 3. Basic Hypergeometric Functions 4. Hypergeometric Integrals 5. Basic Hypergeometric Integrals 6. Bilateral Series 7. Basic Bilateral Series 8. Appell Series 9. Basic Appell Series.

Generalized Hypergeometric Functions. By Lucy Joan Slater. Pp. xiii, 273. 70s. 1966. (Cambridge University Press.)

We define four functions F1∗,F 2∗,F 3∗ and F4∗ as finite field analogues of Appell series F1,F2,F3 and F4, respectively using purely Gauss sums in the spirit of finite field hypergeometric series i...

Appell’s hypergeometric series over finite fields

In this article we find finite field analogues of certain product formulas satisfied by the classical hypergeometric series. We express product of two $${_2}F_1$$
 -Gaussian hypergeometric series as $${_4}F_3$$
 - and $${_3}F_2$$
 -Gaussian hypergeometric series. We use properties of Gauss and Jacobi sums and our earlier works on finite field Appell series to deduce these product formulas satisfied by the Gaussian hypergeometric series. We then use these transformations to evaluate explicitly some special values of $${_4}F_3$$
 - and $${_3}F_2$$
 -Gaussian hypergeometric series. By counting points on CM elliptic curves over finite fields, Ono found certain special values of $${_2}F_1$$
 - and $${_3}F_2$$
 -Gaussian hypergeometric series containing trivial and quadratic characters as parameters. Later, Evans and Greene found special values of certain $${_3}F_2$$
 -Gaussian hypergeometric series containing arbitrary characters as parameters from where some of the values obtained by Ono follow as special cases. We show that some of the results of Evans and Greene follow from our product formulas including a finite field analogue of the classical Clausen’s identity.

Certain product formulas and values of Gaussian hypergeometric series

In this article, we find finite field analogues of certain identities satisfied by the classical $${_4}F_3$$
 -hypergeometric series and Appell series. As an application, we find new summation and product formulas satisfied by the Gaussian hypergeometric series. For example, we express a $${_4}F_3$$
 -Gaussian hypergeometric series as a sum of two $${_2}F_1$$
 -Gaussian hypergeometric series. We also find two identities expressing $${_4}F_3$$
 -Gaussian hypergeometric series as a product of two $${_2}F_1$$
 -Gaussian hypergeometric series. As another application, we find a special value of a $${_4}F_3$$
 -Gaussian hypergeometric series using the summation formula.

Appell series over finite fields and Gaussian hypergeometric series

Many product formulas are known classically for generalized hypergeometric functions over the complex numbers. In this paper, we establish some analogous formulas for generalized hypergeometric functions over finite fields. 

Product formulas for hypergeometric functions over finite fields

We define a function $$F_4^{*}$$
 as a finite field analogue of the classical Appell series $$F_4$$
 using Gauss sums. We establish identities for $$F_4^{*}$$
 analogous to those satisfied by the classical Appell series $$F_4$$
 .

A finite field analogue of the Appell series $F_4$

In this article we find finite field analogues of certain transformations satisfied by the classical hypergeometric series. Using properties of Gauss and Jacobi sums we evaluate certain character sums to establish these transformations. We then use these transformations to evaluate explicitly several special values of $${_2}F_1$$
 hypergeometric functions over finite fields. Certain special values of $${_2}F_1$$
 hypergeometric functions over finite fields containing trivial and quadratic characters obtained by Ono follow from our special values of $${_2}F_1$$
 hypergeometric functions containing arbitrary characters as parameters. One of the finite field analogues of algebraic hypergeometric identities given by Fuselier, Long, Ramakrishna, Swisher, and Tu also follows from one of our transformation formulas.

Mohit Tripathi

Papers

A finite field analogue of the Appell series $F_4$

Appell’s hypergeometric series over finite fields

Certain product formulas and values of Gaussian hypergeometric series

Appell series over finite fields and Gaussian hypergeometric series

Certain transformations and special values of hypergeometric functions over finite fields