By Anatoly N. Kochubei

Dedicated to opposite numbers of classical constructions of mathematical research in research over neighborhood fields of confident attribute, this e-book treats confident attribute phenomena from an analytic point of view. development at the uncomplicated items brought through L. Carlitz - similar to the Carlitz factorials, exponential and logarithm, and the orthonormal method of Carlitz polynomials - the writer develops one of those differential and vital calculi.

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Extra resources for Analysis in Positive Characteristic

Sample text

80) an u(t) = Dn n=0 44 Chapter 1 The norm in H is given by u H = sup |an |. 80) deﬁnes a holomorphic function for n tq |t| < q −1/(q−1) . It is obvious that the sequence of functions f˜n (t) = , Dn n = 0, 1, 2, . , is an orthonormal basis of H. The desired representation is given by the following operators on the space H: a ˜+ = τ, a ˜− = d. Note that the form of the operators a ˜± is only slightly diﬀerent from that ± of a , but they act on a diﬀerent Banach space. 76) hold for the operators a ˜± , with f˜n substituted for fn .

43) j+l=i i gj (t)Gl (s). 44) (iii) If 0 ≤ l < q ν , k ≥ 0, ν ∈ N, then Gk (m)gl (m) = m∈Fq [x] deg m<ν 0, if k + l = q ν − 1, (−1)ν , if k + l = q ν − 1. 0, if k + l = q ν − 1, (−1)ν , if k + l = q ν − 1. 46) Proof. 42) follows from the congruence n−1 n−1 αj ≡ j=0 (mod q − 1). αj q j j=0 Similarly we get the second equality if αj < q−1 for all j. If some αj = q−1, then for that j we get gαj qj (ξt) = gαj qj (t), if ξ = 0. Therefore we come to the required equality assuming that ξ = 0. 43), note that αj αj fj (t + s) αj = (fj (t) + fj (s)) = l=0 αj fj (t)l fj (s)αj −l , l 26 Chapter 1 whence n−1 αj Gj (t + s) = j=0 lj =0 αj fj (t)lj fj (s)αj −lj lj αn−1 α0 ··· = l0 =0 ln−1 =0 αn−1 α0 ··· f0 (t)l0 · · · fn−1 (t)ln−1 l0 ln−1 × f0 (s)α0 −l0 · · · fn−1 (s)αn−1 −ln−1 αn−1 α0 ··· = l0 =0 ln−1 =0 α0 αn−1 Gβ (t)Gi−β (s), ··· l0 ln−1 with β = l0 + l1 q + · · · + ln−1 q n−1 .

15) of the additive Carlitz polynomials were proved by Carlitz in the seminal paper [22] where the polynomials were introduced for the ﬁrst time. Our proofs follow [45]. 16) is due to Wagner [119]. 8) were ﬁrst studied by Wagner [118, 119]. There are several diﬀerent proofs of this result [118, 119, 43, 61, 29, 32]; we followed [119]. 10 was proved in [64]. Hyperdiﬀerentiations were introduced by Hasse [48] and studied in a more general context in [108, 49, 99]; for various generalizations see [112] and references therein.