Determinant of $mathbb F_p$-hypergeometric solutions under ample reduction

We consider the KZ differential equations over $mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $mathbb F_p$. We study the polynomial solutions of these differential equations over $mathbb F_p$, constructed in a previous work joint with V.,Schechtman and called the $mathbb F_p$-hypergeometric solutions. The dimension of the space of $mathbb F_p$-hypergeometric solutions depends on the prime number $p$. We say that the KZ equations have ample reduction for a prime $p$, if the dimension of the space of $mathbb F_p$-hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over $mathbb C$. Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis $mathbb F_p$-hypergeometric solutions. The formula is analogous to the corresponding formula for the determinant of the matrix of coordinates of basis complex hypergeometric solutions, in which binomials $(z_i-z_j)^{M_i+M_j}$ are replaced with $(z_i-z_j)^{M_i+M_j-p}$ and the Euler gamma function $Gamma(x)$ is replaced with a suitable $mathbb F_p$-analog $Gamma_{mathbb F_p}(x)$ defined on $mathbb F_p$.