"Presents a theory of difference and functional equations with continuous argument based on a generalization of the Riemann integral introduced by N. E. Norlund, allowing differentation with respect to the independent variable and permitting greater flexibility in constructing solutions and approximations. Discusses linear transformations that state conditions for convergence of Newton series and Norlund sums!"

David L. Jagerman is a Mathematical Consultant at RUTCOR - Rutgers Center for Operations Research, Rutgers University, Piscataway, New Jersey.

Preface

iii

Operators and Functions

1

(20)

Operators

1

(3)

Factorial function---Stirling numbers

4

(3)

Beta function---factorial Series

7

(3)

Ψ Function and primitives

10

(2)

Laplace and Mellin transformations

12

(5)

Some operational formulae

17

(1)

Problems

17

(4)

Generalities On Difference Equations

21

(11)

Genesis of difference equations

21

(3)

The M/M/C blocking model

24

(1)

The M/M/1 delay model

24

(1)

The time homogeneous first-order model

25

(1)

The Euler equation

25

(6)

Problems

31

(1)

Norlund Sum: Part One

32

(77)

Introduction

32

(1)

Principal solution

33

(1)

Some properties of the sum

34

(3)

Summation of series

37

(1)

Summation by parts

38

(1)

Differentiation

39

(1)

Extension of definition of sum

39

(1)

Repeated summation

40

(1)

Sum of Laplace transforms

41

(3)

Homogeneous form and bounds

44

(6)

Bernoulli's polynomials

50

(6)

Computational formulae

56

(12)

Problems

68

(7)

Norlund Sum: Part Two

Introduction

75

(1)

The Euler-Maclaurin expansion

75

(2)

Existence of the principal sum

77

(7)

Trigonometric expansions

84

(5)

A class of linear transformations

89

(7)

Applications to expansions and functional equations

96

(3)

Application to the Norlund sum

99

(2)

Bound, error estimate, and convolution form

101

(1)

Consideration of some integral equations

102

(3)

Bandlimited functions

105

(1)

Problems

105

(4)

The First-Order Difference Equation

109

(63)

Introduction

109

(1)

The linear homogeneous equation

110

(4)

The inhomogeneous equation

114

(8)

The differential-difference equation

122

(7)

Derivative

129

(3)

Functional equations

132

(6)

U-operator solution of hΔZ=θ(Z)

138

(5)

Critical points

143

(2)

A branching process approximation

145

(2)

A perturbation solution of hΔZ=θ(Z)

147

(2)

Haldane's method for hΔZ=hθ(Z)

149

(2)

Solution of G(φ(z))-l(hz)G(z) = m(z)

151

(3)

Simultaneous first-order equations

154

(8)

Problems

162

(10)

The Linear Equation With Constant Coefficients

172

(28)

Introduction

172

(1)

The homogeneous equation

173

(6)

The inhomogeneous equation

179

(9)

Equations reducible to constant coefficients

188

(1)

Partial difference equations

189

(9)

Problems

198

(2)

Linear Difference Equations With Polynomial Coefficients

200

(37)

Introduction

200

(1)

Depression of order

201

(2)

The operators &pi and ρ

203

(7)

General operational solution

210

(7)

Exceptional cases

217

(5)

The complete equation

222

(4)

The LCFS M/M/C queue with reneging---introduction

226

(2)

Formulation and solution

228

(6)

An M/M/1 processor-sharing queue

234

(2)

Problems

236

(1)

Reference

237

(4)

Index

241

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