复域差分与差分方程

Contents
Preface List of Symbols Chapter 1 Basic Properties of Complex Di.erences 1
Preliminaries 1
1.1
1.2 Di.erence Analogue of the Lemma on the Logarithmic Derivative 5
1.3 Estimations of N (r, f (z +η)) 11
1.4 Estimations of T (r, f (z +η)) 16
1.5 Di.erence Analogue of the Clunie Lemma 17
1.6 Di.erence Analogue of the Mohon’ko Lemma 19
1.7 Di.erence Analogue of the Second Main Theorem 21
nf 27
1.8 Characteristic Functions of Δ
Chapter 2 Relationships Between Meromorphic Functions and Di.erences 59
2.1 Relationships of f (z)and Δnf (z) 59
2.2 Fixed Points of f (z),f (z +c)and Δf (z) 66
Chapter 3 Di.erences of Meromorphic Functions with Small Growth 76
3.1 Divided Di.erences of Entire Functions of order < 1 76
3.2 Di.erences of Entire Functions of order : 1 88
3.3 Di.erences of Meromorphic Functions 90
3.4 Second Di.erences of Meromorphic Functions with Small Growth 100 Chapter 4 Properties of Di.erence Polynomials 115
4.1 Di.erence Analogues of f (z)nf '(z) 115
4.2 Di.erence Analogues of f '(z). af(z)n 128
4.3 Di.erence Analogue of the Tumura-Clunie Type Theorems 135
Chapter 5 Linear First Order Di.erence Equations 140
5.1 Equations Concerning Gamma Function 140
5.2 General Linear First Order Di.erence Equations 153
5.3 Di.erences of Solutions for Di.erence Equations, and Di.erences of Γ(z) 160
Chapter 6 Linear Higher Order Di.erence Equations 173
6.1 Di.erence Equations with Rational Coe.cients 173
6.2 Di.erence Equations with Transcendental Coe.cients 182 Chapter 7 Basic Properties of Nonlinear Di.erence Equations 186
7.1 Basic Properties Concerning Order Number 186
7.2 Properties of Meromorphic Solutions 190
Chapter 8 Di.erence Riccati Equations 199
8.1 Pielou Logistic Equation 199
8.2 Di.erence Riccati Equations 209
8.3 A Family of Solutions for Di.erence Riccati Equations 219
Chapter 9 Di.erence Painlev′eEquations 228
9.1 Classi.edofSecondDi.erenceEquations 228
9.2 Di.erencePainlev′eIIEquations 240
9.3 Di.erence Painlev′eI Equations 251
9.4 Di.erencePainlev′eIIIEquations 258 Chapter 10 Complex q-Di.erences and q-Di.erence Equations 279
10.1 Properties of q-Di.erences 279
10.2 Linear q-Di.erence Equations 288
10.3 Non-Linear q-Di.erence Equations 301
10.4 q-Di.erence Riccati equations 311
Chapter 11 Unicity of Meromorphic Functions Concerning Their Di.erences and Shifts 323
11.1 Meromorphic Functions f (z) that Share Values with f (z +c) 323
11.2 Meromorphic Functions f (z) that Share Values with Δf (z) 330
11.3 Meromorphic Functions f (z) that Share Values with Δf (z)and Δ2f (z) 339
11.4 Meromorphic Functions f (z) that Share Values with Δnf (z) 343
Bibliography 357
Index 368