The common limit of a quadruple sequence and the hypergeometric function $F_{D}$ of three variables

Abstract

We study a quadruple sequence and express its common limit by Lauricella's hypergeometric function $F_{D}(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, 1; z_{1}, z_{2}, z_{3})$ of three variables. We give a functional equation of $F_{D}$, which is the key to get our expression of the common limit.