Jacobi sum, differential equation modulo prime and logarithmic growth

We study the function Fd,m,ν=∫xν(1−xm)ddx for m∈Nm∈N, d∈Qd∈Q and 0⩽ν⩽m−10⩽ν⩽m−1. We regard Fd,m,νFd,m,ν as a p  -adic analytic function. We prove that Fd,m,νFd,m,ν satisfies a Frobenius equation and see that Jacobi sum appears in this equation. We consider the relation between the dimension dimkerbddp(L) of space of bounded solutions of L(y)=(D−νx+dmxm−11−xm)D(y)=0 and the dimension dimKerpL of the space of solutions of L   in Fp[[x]]Fp[[x]]. We prove that dimkerbddp(L)⩽dimKerpL for almost all primes and that dimkerbddp(L)=2 for almost all primes if and only if dimKerpL=2 for almost all primes. Moreover in this case Fd,m,νFd,m,ν is an algebraic function.