Asian Journal of Mathematics

Volume 21 (2017)

Number 5

Special function identities from superelliptic Kummer varieties

Pages: 909 – 952



Adrian Clingher (Department of Mathematics and Computer Science, University of Missouri, St. Louis, Mo., U.S.A.)

Charles F. Doran (Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada; and Department of Physics, University of Maryland, College Park, Md., U.S.A.)

Andreas Malmendier (Department of Mathematics and Statistics, Utah State University, Logan, Ut., U.S.A.)


We prove that a well-known reduction formula by Barnes and Bailey that implies the factorization of Appell’s generalized hypergeometric series into a product of two Gauss’ hypergeometric functions follows entirely from geometry: we first construct a surface of general type as minimal nonsingular model for a product-quotient surface with only rational double points from a pair of superelliptic curves of genus $2r-1$ with $r \in \mathbb{N}$. We then show that this generalized Kummer variety is equipped with two fibrations with general fiber of genus $2r-1$. When periods of a holomorphic two-form over carefully crafted transcendental two-cycles on the generalized Kummer variety are evaluated using either of the two fibrations, the answer must be independent of the fibration and the aforementioned family of special function identities is obtained. This family of identities is a multivariate generalization of Clausen’s Formula. Interestingly, this paper’s finding bridges Ernst Kummer’s two independent lines of research, algebraic transformations for the Gauss’ hypergeometric function and nodal surfaces of degree four in $\mathbb{P}^3$.


Kummer surfaces, product-quotient surfaces, special function identities, hypergeometric functions

2010 Mathematics Subject Classification

14Dxx, 14J28, 33C65

Received 11 April 2016

Accepted 22 April 2016

Published 9 February 2018