Solutions in Terms of Integrals of Multivalued Functions for the Classical Hypergeometric Equations and the Hypergeometric System on the Configuration Space

Abstract

Abstract. The systems of differential equations associated with the classical hypergeometric functions and the hypergeometric functions on the space of point configurations are investigated from the viewpoint of the twisted de Rham theory. In each case, it is proved that the integral of a certain multivalued function over an arbitrary twisted (or loaded) cycle satisfies the system of differential equations in question. The classical hypergeometric functions studied here include Appell's hypergeometric functions IntroductionIn this paper, we investigate the systems of differential equations associated with the classical hypergeometric functions, as well as the hypergeometric functions on the space of point configurations, in the framework of the twisted de Rham theory. In each case, we prove that the integral of a certain multivalued function over an arbitrary twisted (or loaded) cycle satisfies the system of differential equations in question. The classical hypergeometric functions studied here include Appell's hypergeometric functions F 1 , F 2 , F 3 , F 4 , Lauricella's hypergeometric functions F A , F B , F C , F D , and the generalized hypergeometric function n+1 F n . The corresponding differential equations will be denoted by E 1 , E 2 , E 3 , E 4 , E A , E B , E C , E D , and n+1 E n .We briefly explain the point of our investigation, taking the example of integrals associated with n+1 F n :

From positive geometries to a coaction on hypergeometric functions

“…( 7.26) can be written in terms of Appell F 2 functions. This is true, as shown for example in [55]. We now present a brief argument for why this must be the case.…”

“…That this should be possible in the first place can be understood to be a consequence of the fact that each of the Appell functions can be defined by a particular system of linear differential equations (see section 7.3). A proof of this statement in twisted de Rham theory was given in [55] for the Lauricella functions F A , F B , F C , F D , which include the Appell functions as special cases, and for p+1 F p . A suitable choice of bases of forms and cycles will be discussed in each example in turn.…”

“…We finish by noting that all integrals in the period matrix P ( γ; ϕ; Φ), and thus in the coaction eq. (8.10), can be written in terms of p+1 F p functions [55]. For instance, for p = 2 this can be done with the relation…”