Difference Equation for Quintic 3-Fold

Abstract

In this paper, we use the Mellin-Barnes-Watson method to relate solutions of a certain type of 𝑞-difference equations at 𝑄 = 0 and 𝑄 = ∞. We consider two special cases; the first is the 𝑞-difference equation of the 𝛫-theoretic 𝐼-function of the quintic, which is degree 25; we use Adams' method to find the extra 20 solutions at 𝑄 = 0. The second special case is a Fuchsian case, which is confluent to the differential equation of the cohomological 𝐼-function of the quintic. We compute the connection matrix and study the confluence of the 𝑞-difference structure.