An introduction to the calculus of finite differences and difference equations

CONTENTS

Chapter 1. THE CALCULUS OF FINITE DIFFERENCES
1.1. Introduction, 1
1.2. The difference calculus, 2
1.3. Factorial polynomials, 5
1.4. Stirling numbers, 11
1.5. Newton's formula, 18
1.6. The indefinite sum, 21
1.7. The definite sum, 24
1.8. Exercises, 29

Chapter 2. INFINITE PRODUCTS
2.1. Introduction, 36
2.2. Infinite products, 38
2.3. The associated logarithmic series, 42
2.4. Absolute convergence, 44
2.5. Infinite products of functions, 47
2.6. The infinite product representation of the sine function, 49
2.7. The Gamma function, 55
2.8. The Beta function, 60
2.9. The infinite product representation of the Gamma function, 65
2.10. Finite differences and the Gamma function, 73
2.11. Exercises, 77

Chapter 3. BERNOULLI NUMBERS AND POLYNOMIALS
3.1. Introduction, 82
3.2. Generating function for the Bernoulli polynomials, 85
3.3. The Bernoulli numbers, 87
3.4. Properties of the Bernoulli polynomials, 91
3.5. Further properties of the Bernoulli functions, 95
3.6. Power series expansion for tangent and cotangent, 100
3.7. The Euler-Maclaurin formula. Preliminary remarks, 102
3.8. Derivation of the Euler-Maclaurin formula, 105
3.9. Asymptotic expansions, 110
3.10. An application of the Euler-Maclaurin formula, 113
3.11. Stirling's formula, 115
3.12. The algebra of operators, 119
3.13. Exercises, 122

Chapter 4. LINEAR DIFFERENCE EQUATIONS IN THE REAL DOMAIN
4.1. Introduction, 126
4.2. Special formulas, 129
4.3. Linear difference equations, 133
4.4. The nonhomogeneous equation, 143
4.5. Further comments on linear equations, 149
4.6. Linear equations with constant coefficients, 152
4.7. Exercises, 158

References, 163

Index, 165