On certain generating functions in positive characteristic (Q284083)

The second author studied in [Math. Z. 278, No. 1--2, 279--299 (2014; Zbl 1306.11070)] a class of generating functions. In this article the authors present new methods for the study of these generating functions which carry some formal similarities with the Hurwitz zeta function. \textit{M. Papanikolas} introduced in [Invent. Math. 171, No. 1, 123--174 (2008; Zbl 1235.11074)] certain deformations of the Carlitz logarithm. Here the authors prove functional identities (Theorem 2) that establish an explicit connection with the deformations studied by Papanikolas. They find a zeta realization of Papanikola's functions (Theorem 1). The functional identities collect functional identities for some families of the \(L\)-functions introduced by the first author in [Ann. Math. (2) 176, No. 3, 2055--2093 (2012; Zbl 1336.11064)]. The paper also deals with specializations at roots of unity of the generating functions studied here (Theorems 4 and 5). The proofs of the main results are given in Sections 4.1, 4.2 and 5.1.