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17.2. Linear Difference Equations with a Single Continuous Variable . . . . . . . . . . . . . . . . . . 885
17.2.1. First-Order Linear Difference Equations . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 885
17.2.2. Second-Order Linear Difference Equations with Integer Differences . . . . . . . . 894
17.2.3. Linear mth-Order Difference Equations with Integer Differences . . . . . . . . . . 898
17.2.4. Linear mth-Order Difference Equations with Arbitrary Differences .. .. .. .. 904
17.3. Linear Functional Equations .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 907
17.3.1. Iterations of Functions and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 907
17.3.2. Linear Homogeneous Functional Equations .. .. .. .. ... .. .. .. .. .. .. .. .. 910
17.3.3. Linear Nonhomogeneous Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . 912
17.3.4. Linear Functional Equations Reducible to Linear Difference Equations with
Constant Coefficients .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 916
17.4. Nonlinear Difference and Functional Equations with a Single Variable . . . . . . . . . . . . . 918
17.4.1. Nonlinear Difference Equations with a Single Variable . . . . . . . . . . . . . . . . . . 918
17.4.2. Reciprocal (Cyclic) Functional Equations .. .. .. .. .. ... .. .. .. .. .. .. .. .. 919
17.4.3. Nonlinear Functional Equations Reducible to Difference Equations . . . . . . . . 921
17.4.4. Power Series Solution of Nonlinear Functional Equations . . . . . . . . . . . . . . . . 922
17.5. Functional Equations with Several Variables . . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 922
17.5.1. Method of Differentiation in a Parameter . . . . . . . . . . ... .. .. .. .. .. .. .. .. 922
17.5.2. Method of Differentiation in Independent Variables . . . . . . . . . . . . . . . . . . . . . 925
17.5.3. Method of Substituting Particular Values of Independent Arguments . . . . . . . 926
17.5.4. Method of Argument Elimination by Test Functions . ... .. .. .. .. .. .. .. .. 928
17.5.5. Bilinear Functional Equations and Nonlinear Functional Equations Reducible
to Bilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 930
References for Chapter 17 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 935

18. Special Functions and Their Properties .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 937
18.1. Some Coefficients, Symbols, and Numbers .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 937
18.1.1. Binomial Coefficients .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 937
18.1.2. Pochhammer Symbol .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 938
18.1.3. Bernoulli Numbers . .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 938
18.1.4. Euler Numbers . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 939
18.2. Error Functions. Exponential and Logarithmic Integrals . . . . . . . . . . . . . . . . . . . . . . . . 939
18.2.1. Error Function and Complementary Error Function . . ... .. .. .. .. .. .. .. .. 939
18.2.2. Exponential Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 940
18.2.3. Logarithmic Integral . .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 941
18.3. SineIntegral and Cosine Integral. Fresnel Integrals . .. .. .. .. ... .. .. .. .. .. .. .. .. 941
18.3.1. Sine Integral . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 941
18.3.2. Cosine Integral . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 942
18.3.3. Fresnel Integrals . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 942
18.4. Gamma Function, PsiFunction, and BetaFunction . .. .. .. .. ... .. .. .. .. .. .. .. .. 943
18.4.1. Gamma Function . .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 943
18.4.2. Psi Function (Digamma Function) .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 944
18.4.3. Beta Function . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 945
18.5. Incomplete Gamma and BetaFunctions .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 946
18.5.1. Incomplete Gamma Function . .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 946
18.5.2. Incomplete Beta Function . .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 947