By Anatoly N. Kochubei

Dedicated to opposite numbers of classical buildings of mathematical research in research over neighborhood fields of confident attribute, this publication treats confident attribute phenomena from an analytic standpoint. construction at the easy items brought by means of L. Carlitz - equivalent to the Carlitz factorials, exponential and logarithm, and the orthonormal approach of Carlitz polynomials - the writer develops one of those differential and essential calculi.

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45). 1(4) of Sect. 50) that the sequence {Gi }∞ 0 forms an A-basis of the A-module Int(A) of all 30 Chapter 1 integer-valued polynomials from A[t]. 50) in this situation too. It is well known (see, for example, [77], p. 51) if 0 ≤ n ≤ p − 1. 52) if 0 ≤ n ≤ q ν − 1. 14. 16 The polynomials Gqν −1 and gqν −1 satisfy the following identities: (−1)j Gj (t)Gl (s); Gqν −1 (t + s) = j+l=q ν −1 Gqν −1 (t − s) = Gj (t)Gl (s); j+l=q ν −1 (−1)j Gj (t)gl (s); gqν −1 (t + s) = j+l=q ν −1 gqν −1 (t − s) = Gj (t)gl (s).

61) converges if |z| < 1. Below it will be considered for z ∈ K c , |z| < q −1/(q−1) . 23 The Fq -linear functions eC and logC are inverse to each other. Proof. We note ﬁrst of all that eC and logC are mutually inverse as formal power series. Indeed, we have to prove that l (−1)l−n = n Dn Lql−n n=0 l (−1)n = n L Dq n=0 n l−n 0, for l > 0; 1, for l = 0, 0, for l > 0; 1, for l = 0. 15). 22. 61) and the ultra-metric inequality, n |logC (z)| ≤ sup q n |z|q . n≥0 The function ψz (s) = s|z| decreases for s > −(log |z|)−1 ; if |z| < q −1/(q−1) , q−1 n then ψz decreases for s > log q .

Further constructions of orthonormal bases in C0 (O, K) are provided by an appropriate version of the umbral calculus; see Chapter 2 below. The global ﬁeld interpretation of the Carlitz polynomials was initiated by Wagner [118]; see also [43, 29]. 21 are taken from [29] where the case of hyperdiﬀerentiations is also considered in detail. Note that the Carlitz expansions at ﬁnite places have been used by Goss [44] to study, in the spirit of the Iwasawa isomorphism, Oπ -valued measures on Oπ . 6.