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E-raamat: Methods for the Summation of Series [Taylor & Francis e-raamat]

(Illinois Wesleyan University, Bloomington, Illinois, USA)
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Methods for the Summation of SeriesSuurem pilt
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This book presents methods for the summation of infinite and finite series and the related identities and inversion relations. The summation includes the column sums and row sums of lower triangular matrices. The convergence of the summation of infinite series is considered.

The authors focus is on symbolic methods and the Riordan array approach. In addition, this book contains hundreds summation formulas and identities, which can be used as a handbook for people working in computer science, applied mathematics, and computational mathematics, particularly, combinatorics, computational discrete mathematics, and computational number theory. The exercises at the end of each chapter help deepen understanding.

Much of the materials in this book has never appeared before in textbook form. This book can be used as a suitable textbook for advanced courses for high lever undergraduate and lower lever graduate students. It is also an introductory self-study book for re- searchers interested in this field, while some materials of the book can be used as a portal for further research.
Foreword xi
Testimonial xiii
Preface xv
Biography xix
Symbols xxi
1 Classical Methods from Infinitesimal Calculus
1(66)
1.1 Use of Infinitesimal Calculus
2(18)
1.1.1 Convergence of series
2(6)
1.1.2 Limits of sequences and series
8(12)
1.2 Abel's Summation
20(15)
1.2.1 Abel's theorem and Tauber theorem
20(7)
1.2.2 Abel's summation method
27(8)
1.3 Series Method
35(32)
1.3.1 Use of the calculus of finite difference
36(12)
1.3.2 Application of Euler-Maclaurin formula and the Bernoulli polynomials
48(19)
2 Symbolic Methods
67(70)
2.1 Symbolic Approach to Summation Formulas of Power Series
68(28)
2.1.1 Wellknown symbolic expressions
69(5)
2.1.2 Summation formulas related to the operator (1 - xE)-1
74(3)
2.1.3 Consequences and examples
77(5)
2.1.4 Remainders of summation formulas
82(4)
2.1.5 Q-analog of symbolic operators
86(10)
2.2 Series Transformation
96(16)
2.2.1 An extension of Eulerian fractions
98(1)
2.2.2 Series-transformation formulas
99(6)
2.2.3 Illustrative examples
105(7)
2.3 Summation of Operators
112(25)
2.3.1 Summation formulas involving operators
112(8)
2.3.2 Some special convolved polynomial sums
120(3)
2.3.3 Convolution of polynomials and two types of summations
123(2)
2.3.4 Multifold Convolutions
125(5)
2.3.5 Some operator summation formulas from multifold convolutions
130(7)
3 Source Formulas for Symbolic Methods
137(56)
3.1 An Application of Mullin-Rota's Theory of Binomial Enumeration
138(18)
3.1.1 A substitution rule and its scope of applications
138(3)
3.1.2 Various symbolic operational formulas
141(2)
3.1.3 Some theorems on Convergence
143(7)
3.1.4 Examples
150(6)
3.2 On a Pair of Operator Series Expansions Implying a Variety of Summation Formulas
156(22)
3.2.1 A pair of (∞4) degree formulas
157(3)
3.2.2 Specializations and examples
160(8)
3.2.3 A further investigation of a source formula
168(3)
3.2.4 Various consequences of the source formula
171(6)
3.2.5 Lifting process and formula chains
177(1)
3.3 (ΣΔD) General Source Formula and Its Applications
178(15)
3.3.1 (ΣΔD) GSF
178(4)
3.3.2 GSF implies SF(2) and SF(3)
182(1)
3.3.3 Embedding techniques and remarks
183(10)
4 Methods of Using Special Function Sequences, Number Sequences, and Riordan Arrays
193(112)
4.1 Use of Stirling Numbers, Generalized Stirling Numbers, and Eulerian Numbers
194(15)
4.1.1 Basic convergence theorem
194(6)
4.1.2 Summation formulas involving Stirling numbers, Bernoulli numbers, and Fibonacci numbers
200(5)
4.1.3 Summation formulas involving generalized Eulerian functions
205(4)
4.2 Summation of Series Involving Other Famous Number Sequences
209(26)
4.2.1 Convergence theorem and examples
210(6)
4.2.2 More summation formulas involving Fibonacci numbers and generalized Stirling numbers
216(9)
4.2.3 Summation formulas of (ΣS) class
225(10)
4.3 Summation Formulas Related to Riordan Arrays
235(70)
4.3.1 Riordan arrays, the Riordan group, and their sequence characterizations
236(11)
4.3.2 Identities generated by using extended Riordan arrays and Faa di Bruno's formula
247(16)
4.3.3 Various row sums of Riordan arrays
263(9)
4.3.4 Identities generated by using improper or non-regular Riordan arrays
272(7)
4.3.5 Identities related to recursive sequences of order 2 and Girard-Waring identities
279(6)
4.3.6 Summation formulas related to Fuss-Catalan numbers
285(10)
4.3.7 One-pth Riordan arrays and Andrews' identities
295(10)
5 Extension Methods
305(112)
5.1 Identities and Inverse Relations Related to Generalized Riordan Arrays and Sheffer Polynomial Sequences
306(48)
5.1.1 Generalized Riordan arrays and the recurrence relations of their entries
308(10)
5.1.2 The Sheffer group and the Riordan group
318(11)
5.1.3 Higher dimensional extension of the Riordan group
329(13)
5.1.4 Dual sequences and pseudo-Riordan involutions
342(12)
5.2 On an Extension of Riordan Array and Its Application in the Construction of Convolution-type and Abel-type Identities
354(30)
5.2.1 Generalized Riordan arrays with respect to basic sequences of polynomials
354(10)
5.2.2 A general class of convolution-type identities
364(10)
5.2.3 A general class of Abel identities
374(10)
5.3 Various Methods for constructing Identities Related to Bernoulli and Euler Polynomials
384(33)
5.3.1 Applications of dual sequences to Bernoulli and Euler polynomials
387(12)
5.3.2 Extended Zeilberger's algorithm for constructing identities related to Bernoulli and Euler polynomials
399(1)
5.3.2.1 Gosper's algorithm
399(3)
5.3.2.2 W-Z algorithm
402(1)
5.3.2.3 Zeilberger's creative telescoping algorithm
403(2)
5.3.2.4 Extended Zeilberger's algorithm
405(12)
Bibliography 417(16)
Index 433
Tian-Xiao He received Ph. D. degrees in at Dalian University of Technology, Dalian, and Texas A&M University, College Station, respectively. Dr. He is a Professor of Mathematics at Illinois Wesleyan University and Earl and Marian A. Beling Professor of Natural Science at Illinois Wesleyan University. Dr. He has authored or co-authored over 150 research articles and 7 volumes in mathematics and is presently an editor/chief editor for several math journals.
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