| 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) |
|
|
|
20 | (15) |
|
1.2.1 Abel's theorem and Tauber theorem |
|
|
20 | (7) |
|
1.2.2 Abel's summation method |
|
|
27 | (8) |
|
|
|
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) |
|
|
|
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) |
|
|
|
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) |
|
|
|
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) |
|
|
|
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) |
|
|
|
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 | |
