A cubic transformation formula for Appell–Lauricella hypergeometric functions over finite fields

被引：7
作者：
Frechette S. [1 ]
Swisher H. [2 ]
Tu F.-T. [3 ]
机构：
[1] Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, 01610, MA
[2] Department of Mathematics, Oregon State University, Corvallis, 97331, OR
[3] Department of Mathematics, Louisiana State University, Baton Rouge, 70803, LA
来源：
Research in Number Theory | 2018年 / 4卷 / 2期
关键词：
Finite fields; Hypergeometric functions; Lauricella series; Multivariable hypergeometric functions; Picard curves;
D O I：
10.1007/s40993-018-0119-9
中图分类号：
学科分类号：
摘要：
We define a finite field version of Appell–Lauricella hypergeometric functions built from period functions in several variables, paralleling recent development by Fuselier, Long, Ramakrishna, and the last two authors in the single variable case. We develop geometric connections between these functions and the family of generalized Picard curves. In our main result, we use finite field Appell–Lauricella functions to establish a finite field analogue of Koike and Shiga’s cubic transformation for the Appell hypergeometric function F1, proving a conjecture of Long. We use our multivariable period functions to construct formulas for the number of Fp-points on the generalized Picard curves. We also give some transformation and reduction formulas for the period functions, and consequently for the finite field Appell–Lauricella functions. © 2018, SpringerNature.
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