An Introduction to the Calculus of Finite Differences and Difference Equations

Preface v

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1. The Calculus of Finite Differences
...1. Introduction 1
...2. The difference calculus 2
...3. Factorial polynomials 5
...4. Stirling numbers 11
...5. Newton's formula 18
...6. The indefinite sum 21
...7. The definite sum 24
...Exercises 29

2. Infinite Products
...1. Introduction 36
...2. Infinite products 38
...3. The associated logarithmic series 42
...4. Absolute convergence 44
...5. Infinite products of functions 47
...6. The infinite product representation of the sine function 49
...7. The Gamma function 55
...8. The Beta function 60
...9. The infinite product representation of the Gamma function 65
...10. Finite differences and the Gamma function 73
...Exercises 77

3. Bernoulli Numbers and Polynomials
...1. Introduction 82
...2. Generating function for the Bernoulli polynomials 85
...3. The Bernoulli numbers 87
...4. Properties of the Bernoulli polynomials 91
...5. Further properties of the Bernoulli functions 95
...6. Power series expansion for tangent and cotangent 100
...7. The Euler-Maclaurin formula. Preliminary remarks 102
...8. Derivation of the Euler-Maclaurin formula 105
...9. Asymptotic expansions 110
...10. An application of the Euler-Maclaurin formula 113
...11. Stirling's formula 115
...12. The algebra of operators 119
...Exercises 122

4. Linear Difference Equations in the Real Domain
...1. Introduction 126
...2. Special formulas 129
...3. Linear difference equations 133
...4. The nonhomogeneous equation 143
...5. Further comments on linear equations 149
...6. Linear equations with constant coefficients 152
...Exercises 158
...
References 163
Index 165