Devoted to counterparts of classical structures of mathematical analysis in analysis over local fields of positive characteristic, this book treats positive characteristic phenomena from an analytic viewpoint. Building on the basic objects introduced by L. Carlitz - such as the Carlitz factorials, exponential and logarithm, and the orthonormal system of Carlitz polynomials - the author develops a kind of differential and integral calculi.
Author(s): Anatoly N. Kochubei
Series: Cambridge Tracts in Mathematics
Edition: 1
Publisher: Cambridge University Press
Year: 2009
Language: English
Pages: 222
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 9
1.1 Basic notions......Page 13
1.2 Additive Carlitz polynomials......Page 25
1.3 Hyperdifferentiations......Page 29
1.4 The digit principle......Page 34
1.5 Finite places of a global function field......Page 44
1.6 The Carlitz module......Page 48
1.7 Canonical commutation relations......Page 52
1.8 Comments......Page 56
2.1 Fq-Linear calculus......Page 59
2.2 Umbral calculus......Page 72
2.3 Locally analytic functions......Page 87
2.4 General smooth functions......Page 94
2.5 Entire functions......Page 102
2.6 Measures and divided power series......Page 111
3.1 Existence and uniqueness theorems......Page 117
3.2 Strongly nonlinear equations......Page 126
3.3 Regular singularity......Page 132
3.4 Evolution equations......Page 143
3.5 Comments......Page 147
4.1 Hypergeometric functions......Page 149
4.2 Analogs of the Bessel functions and Jacobi polynomials......Page 154
4.3 Polylogarithms......Page 157
4.4 K-binomial coefficients......Page 166
4.5 Overconvergence properties......Page 171
4.6 Comments......Page 178
5.1 Algebraic preliminaries......Page 179
5.2 The Carlitz rings......Page 185
5.3 The ring A1......Page 190
5.4 Quasi-holonomic modules......Page 197
5.5 Comments......Page 213
Bibliography......Page 215
Index......Page 221
