Certain transformations for hypergeometric series in the p-adic setting

Abstract

Abstract. In [12], McCarthy defined a function nGn[· · · ] using the Teichmüller character of finite fields and quotients of the p-adic gamma function. This function extends hypergeometric functions over finite fields to the p-adic setting. In this paper, we give certain transformation formulas for the function nGn[· · · ] which are not implied from the analogous hypergeometric functions over finite fields.

Hypergeometric type identities in the $p$-adic setting and modular forms

“…There are very few identities and transformations for n G n [· · · ] in full generality. Barman and Saikia [4,5] provide transformations for 2 G 2 [· · · ], by counting points on various families of elliptic curves. To our knowledge Theorems 2.4 to 2.7 are the only other full n G n [· · · ] identities.…”

Summation identities and special values of hypergeometric series in the p-adic setting

“…While numerous transformations exist for the finite field hypergeometric functions, very few exist for n G n [· · · ] in full generality. The first and second authors [5,6] provide transformations for 2 G 2 [· · · ] q by counting points on various families of elliptic curves over F q . Recently, the third author and Fuselier [10] provide two more transformations for n G n [· · · ] p when n = 3 and n = 4, respectively.…”

“…While numerous transformations exist for the finite field hypergeometric functions, very few exist for n G n [• • • ] in full generality. The first and second authors [5,6] provide transformations for 2 G 2 [• • • ] q by counting points on various families of elliptic curves over F q .…”

“…Remark 1.8. In [5,6], the first and second authors derived transformations for the function 2 G 2 [• • • ] q with different parameters. We can now derive many such transformations by combining the transformation of Theorem 1.7 with those given in [5,6].…”

“…The technique used to derive the summation identities for n G n [· · · ] q is based on counting points on certain families of hyperelliptic curves and counting zeros on certain families of polynomials in terms of n G n [· · · ] q . We also applied this method successfully to derive many transformations for 2 G 2 [· · · ] q by counting points on various families of elliptic curves (see for example [5,6]). We now pose an open problem below.…”