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

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.

💡 Research Summary

The paper investigates transformation formulas for the multivariate Appell‑Lauricella hypergeometric functions (F_D) by exploiting double integrals and period integrals on algebraic surfaces. The authors begin by recalling the classical quadratic and higher‑order transformations of the Gauss hypergeometric function ({}_2F_1) discovered by Gauss, Kummer and Goursat, emphasizing that such identities often reflect underlying relations between monodromy groups. They then describe the multivariate generalizations, namely the Appell function (F_1) and the full Appell‑Lauricella family (F_D), together with their Euler‑type integral representations.

A central technical contribution is a set of lemmas (Lemma 3.1–3.3) that convert one‑dimensional period integrals of the form (\int_{x_1}^{x_2} (x-x_1)^{\mu_1-1}(x_2-x)^{\mu_2-1}\prod|x-x_i|^{\mu_i},dx) into beta‑function factors and Appell‑Lauricella functions after a suitable Möbius transformation. These lemmas are repeatedly applied to the inner integrals that arise when evaluating a double integral on a surface.

The geometric setting is a surface (S) obtained as a cyclic (d)-fold cover of (\mathbb{P}^1\times\mathbb{P}^1) branched along a divisor (D=\sum_i D_i) of total bidegree ((3,3)). The branching data are encoded by real exponents (\mu_i\in(0,1)) satisfying (\sum_i\mu_i=2). On the complement (U=\mathbb{P}^1\times\mathbb{P}^1\setminus D) a rank‑one local system (L) is defined by the monodromy (e^{2\pi i\mu_i}) around each component. The holomorphic 2‑form \