| Preface Acknowledgments 1. SOME METHODS FOR CLOSED FORM REPRESENTATION 1 Some Methods 1.1 Introduction 1.2 Contour Integration 1.3 Use of Integral Equations 1.4 Wheelon's Results 1.5 Hypergeometric Functions 2 A Tree Search Sum and Some Relations 2.1 Binomial Summation 2.2 Riordan 2.3 Method of Jonassen and Knuth 2.4 Method of Gessel 2.5 Method of Rousseau 2.6 Hypergeometric Form 2.7 Snake Oil Method 2.8 Some Relations 2.9 Method of Sister Celine 2.10 Method of Creative Telescoping 2.11 WZ Pairs Method 2. NON-HYPERGEOMETRIC SUMMATION 1 Introduction 2 Method 3 Bfirmalm's Theorem and Application 4 Differentiation and Integration 5 Forcing Terms 6 Multiple Delays, Mixed and Neutral Equations 7 Bruwier Series 8 Teletraffic Example 9 Neutron Behaviour Example 10 A Renewal Example 11 Ruin Problems in Compound Poisson Processes 12 A Grazing System 13 Zeros of the Transcendental Equation 14 Numerical Examples 15 Euler's Work 16 Jensen's Work 17 Ramanujan's Question 18 Cohen's Modification and Extension 19 Conolly's Problem 3. BURMANN'S THEOREM 1 Introduction 2 Bfirmann's Theorem and Proof 2.1 Applying Biirmann's Theorem 2.2 The Remainder 3 Convergence Region 3.1 Extension of the Series 4. BINOMIAL TYPE SUMS 1 Introduction 2 Problem Statement 3 A Recurrence Relation 4 Relations Between Gk (m) and Fk+l (m) 5. GENERALIZATION OF THE EULER SUM 1 Introduction 2 1-Dominant Zero 2.1 The System 2.2 QR,K (O) Recurrences and Closed Forms 2.3 Lemma and Proof of Theorem 5.1 2.4 Extension of Results …… 6. HYPERGEOMETRIC SUMMATION:FIBONACCI AND RELATED SERIES 7. SUMS AND PRODUCTS OF BINOMIAL TYPE 8. SUMS OF BINOMIAL VARIATION References About the Author Index |
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