Double integrals and transformation formulas for Appell--Lauricella hypergeometric functions $F_D$

TL;DR

The paper develops a geometric framework based on double fibrations and period integrals to derive transformation formulas among Appell–Lauricella hypergeometric functions $F_D^{(n)}$. By encoding divisors on $\mathbb{P}^1\times\mathbb{P}^1$, constructing a cyclic branched cover, and applying two different iterated integrations, the authors produce explicit identities between multivariable hypergeometric functions and their degenerations to Gauss ${}_2F_{1}$. The main results include a suite of transformation formulas for various divisor partitions, with several degenerate cases yielding classical quadratic transformations such as those of Goursat. The methodology provides a period-integral perspective on monodromy and its relation to functional transformations, offering an alternative route to known transformations and new relations for Appell–Lauricella functions.$

Abstract

The monodromy of hypergeometric functions can govern the properties of the functions themselves. Previously, the second and third authors studied the commensurability relations among monodromy groups of the Appell--Lauricella hypergeometric functions using Deligne--Mostow theory and the geometric correspondence between curves and surfaces. In this paper, we apply the same construction to obtain transformation formulas among these hypergeometric functions. This also provides an alternative approach to some of Goursat's quadratic transformations via double integrals and Fubini's theorem.

Figures (9)

• Figure 1: $(2,1)+(1,1)+(0,1)$
• Figure 2: $(1,2)+(1,0)+(1,0)+(0,1)$, Segment Region
• Figure 3: $(1,2)+(1,0)+(1,0)+(0,1)$, Sector Region
• Figure 4: (1,2)+(1,0)+(1,0)+(0,1), Degeneration 1
• Figure 5: (1,2)+(1,0)+(1,0)+(0,1), Degeneration 2

Theorems & Definitions (1)

• proof