Computational Techniques for the Summation of Series

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 Burmann'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'sWork.- 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 Burmann's Theorem and Proof.- 2.1 Applying Burmann'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+1 (m).- 5. Generalization of the Euler Sum.- 1 Introduction.- 2 1-Dominant Zero.- 2.1 The System.- 2.2 QR,k (0) Recurrences and Closed Forms.- 2.3 Lemma and Proof of Theorem 5.1.- 2.4 Extension of Results.- 2.5 Renewal Processes.- 3 The K-Dominant Zeros Case.- 3.1 The k-System.- 3.2 Examples.- 3.3 Extension.- 6. Hypergeometric Summation: Fibonacci and Related Series.- 1 Introduction.- 2 The Difference-Delay System.- 3 The Infinite Sum.- 4 The Lagrange Form.- 5 Central Binomial Coefficients.- 5.1 Related Results.- 6 Fibonacci, Related Polynomials and Products.- 7 Functional Forms.- 7. Sums and Products of Binomial Type.- 1 Introduction.- 2 Technique.- 3 Multiple Zeros.- 4 More Sums.- 5 Other Forcing Terms.- 8. Sums of Binomial Variation.- 1 Introduction.- 2 One Dominant Zero.- 2.1 Recurrences.- 2.2 Proof of Conjecture.- 2.3 Hypergeometric Functions.- 2.4 Forcing Terms.- 2.5 Products of Central Binomial Coefficients.- 3 Multiple Dominant Zeros.- 3.1 The k Theorem.- 4 Zeros.- 4.1 Numerical Results and Special Cases.- 4.2 The Hypergeometric Connection.- 5 Non-zero Forcing Terms.- References.- About the Author.