Explicit formulae for $$L$$ L -values in positive characteristic

Abstract

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.

On twisted A-harmonic sums and Carlitz finite zeta values

“…In [19], the second author found several explicit formulas for the sums ψ s = ψ s,∞ = lim d→∞ ψ s,d for small values of s and in [16] the two authors of the present paper improved qualitatively the previous results without any restriction on s. In particular, the following formula holds, with t = t 1 , valid for d ≥ 1:…”

Functional identities for L-series values in positive characteristic

“…The Conjecture follows from Perkins results [15] in the case s ≤ q and α = s. The conjecture is also verified if ℓ q (s) = q and α = 1, thanks to Corollary 26. …”

“…He notably studied the growth of their degrees. Moreover, by using Wagner's interpolation theory for the map χ t , Perkins [15] obtained explicit formulas for the series…”

Polynomials of binomial type and Lucas’ Theorem

“…The umbral theory of [RT1], with its "black magic" of linear maps etc., has further connections with the arithmetic of function fields as pointed out by F. Pellarin and which we briefly describe here. Let C be the Carlitz module and let ω(t) be the Anderson-Thakur function as in [Pe1]. Let τ be the q-th power mapping acting as in [Pe1] where one defines (Definition 2.6) the polynomials b j (t) by τ j ω(t) = b j (t)ω(t).…”

“…Let C be the Carlitz module and let ω(t) be the Anderson-Thakur function as in [Pe1]. Let τ be the q-th power mapping acting as in [Pe1] where one defines (Definition 2.6) the polynomials b j (t) by τ j ω(t) = b j (t)ω(t). It is readily seen that b j (t) = j−1 e=0 (t − θ q e ) (and in fact, as shown in ibid, these polynomials are universal in that the coefficients of both the Carlitz exponential and logarithm may be easily expressed using them).…”