Umbral Calculus (eBook)

Front Cover 1
The Umbral Calculus 4
Copyright Page 5
Contents 8
Preface 10
CHAPTER 1. INTRODUCTION 12
1. A Definition of the Classical Umbral Calculus 12
2. Preliminaries 14
CHAPTER 2. SHEFFER SEQUENCES 17
1. The Umbral Algebra 17
2. Linear Operators 23
3. Sheffer Sequences 28
4. Associated Sequences 36
5. Appell Sequences 37
6. A Few Examples 39
CHAPTER 3. OPERATORS AND THEIR ADJOINTS 43
1. Continuous Operators on P* 43
2. Automorphisms and Derivations on P* 44
3. Adjoints 45
4. Umbral Operators and Umbral Composition 48
5. Sheffer Operators and Umbral Composition 53
6. Umbral Shifts and the Recurrence Formula for Associated Sequences 56
7. Sheffer Shifts and the Recurrence Formula for Sheffer Sequences 60
8. The Transfer Formulas 61
CHAPTER 4. EXAMPLES 64
1. Associated Sequences 64
2. Appell Sequences 97
3. Sheffer Sequences 118
4. Other Shefi'er Sequences 136
CHAPTER 5. TOPICS 142
1. The Connection Constants Problem and Duplication Formulas 142
2. The Lagrange Inversion Formula 149
3. Cross Sequences and Steffensen Sequences 151
4. Operational Formulas 155
5. Inverse Relations 158
6. Sheffer Sequence Solutions to Recurrence Relations 167
7. Binomial Convolution 171
CHAPTER 6. NONCLASSICAL UMBRAL CALCULI 173
1. Introduction 173
2. The Principal Results 173
3. A Particular Delta Series and the Gegenbauer Polynomials 177
4. The q-Umbra1 Calculus 185
REFERENCES 194
Index 200