Doctor of Philosophy, The Ohio State University, 2013, Mathematics

We consider various methods for explicitly computing the special values of Pellarin's L-series in several indeterminates, both at the positive integers and negative integers. Our premier result is the explicit calculation, in terms of the Goss-Carlitz zeta values, of the rational functions appearing in the paper by Pellarin in which these L-series were first introduced, as well as an explicit generalization of his result to at most q indeterminates. We draw applications to new divisibility results for the numerators of the Bernoulli-Carlitz numbers by degree one irreducible polynomials and to explicit generating series and recursive relations for Pellarin's series. On the negative integers side, we deepen some work of Goss on the special polynomials associated to Pellarin's series. In particular, we show that when certain parameters are fixed, the logarithmic growth in the degrees of the aforementioned special polynomials that is common in this area holds in great generality. We study the natural action of Goss' group of digit permutations on a sub-class of these special polynomials and show that their degrees are an invariant of the action of Goss' group. Finally, we include a computation of the coefficients of the measure whose moments are the special polynomials associated to Pellarin's L-series in one indeterminate. The Wagner series for Pellarin's evaluation character will play a central role in nearly all of the results of this dissertation. We will show that the Wagner series for Pellarin's evaluation character, an object arising in the study of the completions of our ring of integers at the finite places, will also have meaning at the infinite place. In fact, at the infinite place, this Wagner series is none other than Anderson's generating function for the Carlitz module, and we shall see that Pellarin's L-series, the Wagner series for Pellarin's character and Anderson's generating function are inextricably tied

Committee: David Goss (Advisor); Warren Sinnott (Committee Member); Wenzhi Luo (Committee Member) Subjects: Mathematics