## Appell series over finite fields and Gaussian hypergeometric series

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2021

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In this thesis we study classical hypergeometric series and Appell series over finite fields, and find finite field analogues of several product and summation formulas satisfied by the classical hypergeometric series. Hypergeometric functions over finite fields are known as Gaussian hypergeometric series. As an application of the product and summation formulas, we deduce several special vlaues of 2F1, 3F2 and 4F3-Gaussian hypergeometric series. Some of our special values of Gaussian hypergeometric series are evaluated at general arguments of the parameters. Recently, finite field alanogues of Appell series F1, F2 and F3 are introduced and their relations with certain Gaussian hypergeometric series are established. Integral representations of F1, F2 and F3 are used while defining their finite field analogues. However, integral representations of F4 are more complicated than the integral representations of F1, F2 and F3. Therefore, it is not straightforward to find an appropriate finite field analogue of F4 using its integral representations. To overcome this problem, we define finite field analogues of classical Appell series F1, F2 and F3 using purely Gauss sums, and this allows us to define a finite field analogue of the Appell series F4. We then establish finite field analogues of classical identities satisfied by the Appell series and hypergeometric series. As applications, we find several transformation formulas satisfied by the Gaussian hypergeometric series. For example, we express a 4F3-Gaussian hypergeometric series as a sum of Mo 2F1-Gaussian hypergeometric series. We also express 4F3-Gaussian hypergeometric series as a product of two 2F1-Gaussian hypergeometric series. Product formulas for Gaussian hypergeometric series have many significant applications. We find finite field analogues of certain product formulas satistied by the classical hypergeometric series. We express product of two 2F1-Gaussian hypergeometric series as 4F3- and 3F2-Gaussian hypergeometric series. We use properties of Gauss and Jacobi sums and our 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 4F3- and 3F2-Gaussian hypergeometric series. By counting points on CM elliptic curves over finite fields, Ono found certain special values of 2F1- and 3F2-Gaussian hypergeometric series containing trivial and quadratic characters as parameters. Later, Evans and Greene found special values of certain 3F2-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

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Supervisor: Rupam Barman

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MATHEMATICS