An analogue of Girstmair's formula in function fields

Suppose that p is an odd prime and $g>1$ is a primitive root modulo p. Let M be a number field contained in the p-th cyclotomic field. Girstmair found a surprising relation between the relative class number of M and the digits of $1/p$ in base g. In this paper, we consider an analogue of Girstmair's formula in function fields. Suppose that P ∈ Fq[T] is monic irreducible and G ∈ Fq[T] is a primitive root modulo P. Let L be an extension field of Fq(T) contained in the P-th cyclotomic function field. The goal of this paper is to give relations between the plus and minus parts of the divisor class number of L and the digits of $1/P$ in base G.