Methods for the Summation of Series

Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Foreword
Testimonial
Preface
Biography
Symbols
1. Classical Methods from Infinitesimal Calculus
	1.1. Use of Infinitesimal Calculus
		1.1.1. Convergence of series
		1.1.2. Limits of sequences and series
	1.2. Abel’s Summation
		1.2.1. Abel’s theorem and Tauber theorem
		1.2.2. Abel’s summation method
	1.3. Series Method
		1.3.1. Use of the calculus of finite difference
		1.3.2. Application of Euler-Maclaurin formula and the Bernoulli polynomials
2. Symbolic Methods
	2.1. Symbolic Approach to Summation Formulas of Power Series
		2.1.1. Wellknown symbolic expressions
		2.1.2. Summation formulas related to the operator (1 − xE)−1
		2.1.3. Consequences and examples
		2.1.4. Remainders of summation formulas
		2.1.5. Q-analog of symbolic operators
	2.2. Series Transformation
		2.2.1. An extension of Eulerian fractions
		2.2.2. Series-transformation formulas
		2.2.3. Illustrative examples
	2.3. Summation of Operators
		2.3.1. Summation formulas involving operators
		2.3.2. Some special convolved polynomial sums
		2.3.3. Convolution of polynomials and two types of summations
		2.3.4. Multifold Convolutions
		2.3.5. Some operator summation formulas from multifold convolutions
3. Source Formulas for Symbolic Methods
	3.1. An Application of Mullin-Rota’s Theory of Binomial Enumeration
		3.1.1. A substitution rule and its scope of applications
		3.1.2. Various symbolic operational formulas
		3.1.3. Some theorems on Convergence
		3.1.4. Examples
	3.2. On a Pair of Operator Series Expansions Implying a Variety of Summation Formulas
		3.2.1. A pair of (∞4) degree formulas
		3.2.2. Specializations and examples
		3.2.3. A further investigation of a source formula
		3.2.4. Various consequences of the source formula
		3.2.5. Lifting process and formula chains
	3.3. (
D) General Source Formula and Its Applications
		3.3.1. (
D) GSF
		3.3.2. GSF implies SF(2) and SF(3)
		3.3.3. Embedding techniques and remarks
4. Methods of Using Special Function Sequences, Number Sequences, and Riordan Arrays
	4.1. Use of Stirling Numbers, Generalized Stirling Numbers, and Eulerian Numbers
		4.1.1. Basic convergence theorem
		4.1.2. Summation formulas involving Stirling numbers, Bernoulli numbers, and Fibonacci numbers
		4.1.3. Summation formulas involving generalized Eulerian functions
	4.2. Summation of Series Involving Other Famous Number Sequences
		4.2.1. Convergence theorem and examples
		4.2.2. More summation formulas involving Fibonacci numbers and generalized Stirling numbers
		4.2.3. Summation formulas of (S) class
	4.3. Summation Formulas Related to Riordan Arrays
		4.3.1. Riordan arrays, the Riordan group, and their sequence characterizations
		4.3.2. Identities generated by using extended Riordan arrays and Fa`a di Bruno’s formula
		4.3.3. Various row sums of Riordan arrays
		4.3.4. Identities generated by using improper or non-regular Riordan arrays
		4.3.5. Identities related to recursive sequences of order 2 and Girard-Waring identities
		4.3.6. Summation formulas related to Fuss-Catalan numbers
		4.3.7. One-pth Riordan arrays and Andrews’ identities
5. Extension Methods
	5.1. Identities and Inverse Relations Related to Generalized Riordan Arrays and Sheffer Polynomial Sequences
		5.1.1. Generalized Riordan arrays and the recurrence relations of their entries
		5.1.2. The Sheffer group and the Riordan group
		5.1.3. Higher dimensional extension of the Riordan group
		5.1.4. Dual sequences and pseudo-Riordan involutions
	5.2. On an Extension of Riordan Array and Its Application in the Construction of Convolution-type and Abel-type Identities
		5.2.1. Generalized Riordan arrays with respect to basic sequences of polynomials
		5.2.2. A general class of convolution-type identities
		5.2.3. A general class of Abel identities
	5.3. Various Methods for constructing Identities Related to Bernoulli and Euler Polynomials
		5.3.1. Applications of dual sequences to Bernoulli and Euler polynomials
		5.3.2. Extended Zeilberger’s algorithm for constructing identities related to Bernoulli and Euler polynomials
			5.3.2.1. Gosper’s algorithm
			5.3.2.2. W-Z algorithm
			5.3.2.3. Zeilberger’s creative telescoping algorithm
			5.3.2.4. Extended Zeilberger’s algorithm
Bibliography
Index