Explicit formulae for L-values in positive characteristic

We focus on the generating series for the rational special values of Pellarin’s (L) -series in (1 le s le 2(q-1)) indeterminates, and using interpolation polynomials we prove a closed form formula relating this generating series to the Carlitz exponential, the Anderson–Thakur function, and the Anderson generating functions for the Carlitz module. We draw several corollaries, including explicit formulae and recursive relations for Pellarin’s (L) -series in the same range of (s) , and divisibility results on the numerators of the Bernoulli–Carlitz numbers by monic irreducibles of degrees one and two.