Abstract. We focus on the generating series for the rational special values of Pellarin's Lseries in 1 ≤ s ≤ 2(q − 1) indeterminates, and using interpolation polynomials we prove a closed form formula relating this generating series to the Carlitz exponential, the AndersonThakur 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.

Necessary and sufficient conditions are given for a negative integer to be a trivial zero of a new type of L-series recently discovered by F. Pellarin, and it is shown that any such trivial zero is simple. We determine the exact degree of the special polynomials associated to Pellarin's L-series. The theory of Carlitz polynomial approximations is developed further for both additive and Fq-linear functions. Using Carlitz' theory we give generating series for the power sums occurring as the coefficients of the special polynomials associated to Pellarin's series, and a connection is made between the Wagner representation for χt and the value of Pellarin's L-series at 1.

We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit connection with certain deformations of the Carlitz logarithm introduced by M. Papanikolas and involve the Anderson-Thakur function and the Carlitz exponential function. They collect certain functional identities in families for a new class of L-functions introduced by the first author. This paper also deals with specializations at roots of unity of these generating functions, producing a link with Gauss-Thakur sums. Positive characteristic and Carlitz module and Anderson generating functions and L-series and Special values and Periodic functions 1

Abstract. In this text, we develop the theory of vectorial modular forms with values in Tate algebras introduced by the first author, in a very special case (dimension two, for a very particular representation of Γ := GL 2 (Fq[θ])). Among several results that we prove here, we determine the complete structure of the modules of these forms, we describe their specializations at roots of unity and their connection with Drinfeld modular forms for congruence subgroups of Γ and we prove that the modules generated by these forms are stable under the actions of Hecke operators.