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基本説明
Presents a unified treatment of summation of sums and series using function theoretic methods.
Full Description
"This book collects in one volume the author's considerable results in the area of the summation of series and their representation in closed form, and details the techniques by which they have been obtained... the calculations are given in plenty of detail, and closely related work which has appeared in a variety of places is conveniently collected together." --The Australian Mathematical Society Gazette
Contents
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. Bürmann's Theorem.- 1 Introduction.- 2 Bürmann's Theorem and Proof.- 2.1 Applying Bürmann'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.
