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• 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