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

Abstract

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 F 1 , proving a conjecture of Long. We use our multivariable period functions to construct formulas for the number of F p -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.

Pathway Fractional Integral Operator on Some τ-extensions of Lauricella Functions of Several Variables

“…et al [35]. Lauricella functions have been transformed as summation formulas [32], decomposition formulas [18], extension formulas [1], reduction formulas [11], a cubic transformation formula [14], an expression over a finite field [10], incomplete Lauricella functions [12], incomplete Lauricella matrix functions [36], and τ-extension of Lauricella functions of several variables [22] with applications on heat equations [19].…”