Foreword ...................................................... vii
Preface ........................................................ ix
Acknowledgments ................................................ xi
1 Basic Properties of Series ................................... 1
1.1 General Considerations of Σnk=α ƒ(k)...................... 4
1.2 Pascal's Identity in Evaluation of Series ............... 8
2 The Binomial Theorem ........................................ 13
2.1 Newton's Binomial Theorem and the Geometric Series ..... 16
3 Iterative Series ............................................ 21
3.1 Two Summation Interchange Formulas ..................... 23
3.2 Gould's Convolution Formula ............................ 28
4 Two of Professor Gould's Favorite Algebraic Techniques ...... 35
4.1 Coefficient Comparison ................................. 35
4.2 The Fundamental Theorem of Algebra ..................... 42
5 Vandermonde Convolution ..................................... 49
5.1 Five Basic Applications of the Vandermonde
Convolution ............................................ 50
5.2 An In-Depth Investigation Involving Equation (5.4) ..... 55
6 The nth Difference Operator and Euler's Finite
Difference Theorem .......................................... 63
6.1 Euler's Finite Difference Theorem ...................... 68
6.2 Applications of Equation (6.16) ........................ 69
7 Melzak's Formula ............................................ 79
7.1 Basic Applications of Melzak's Formula ................. 83
7.2 Two Advanced Applications of Melzak's Formula .......... 86
7.3 Partial Fraction Generalizations of Equation (7.1) ..... 89
7.4 Lagrange Interpolation Theorem ........................ 93
8 Generalized Derivative Formulas ............................ 101
8.1 Leibniz Rule .......................................... 101
8.2 Generalized Chain Rule ................................ 102
8.3 Five Applications of Hoppe's Formula .................. 105
9 Stirling Numbers of the Second Kind S(n, k) ................ 113
9.1 Euler's Formula for S(n, k) ........................... 118
9.2 Grunert's Operational Formula ......................... 126
9.3 Expansions of (℮x-1)n ................................. 131
x
9.4 Bell Numbers .......................................... 133
10 Eulerian Numbers ........................................... 139
10.1 Functional Expansions Involving Eulerian Numbers ...... 142
10.2 Combinatorial Interpretation of A(n, m) ............... 144
11 Worpitzky Numbers .......................................... 147
11.1 Polynomial Expansions from Nielsen's Formula .......... 152
11.2 Nielsen's Expansion with Taylor's Theorem ............. 156
11.3 Nielsen Numbers ....................................... 161
12 Stirling Numbers of the First Kind s(n, k) ................. 165
12.1 Properties of s(n, k) ................................. 168
12.2 Orthogonality Relationships for Stirling Numbers ...... 170
12.3 Functional Expansions Involving s(n, k) ............... 173
12.4 Derivative Expansions Involving s(n, k) ............... 175
13 Explicit Formulas for s(n, n - k) .......................... 177
13.1 Schlдfli's Formula s(n, n - k) ........................ 180
13.2 Proof of Equation (13.2) .............................. 185
14 Number Theoretic Definitions of Stirling Numbers ........... 191
14.1 Relationships Between S1(n, k) and S2(n, k) ........... 196
14.2 Hдgen Recurrences for S1(n, k) and S2(n, k) ........... 199
15 Bernoulli Numbers .......................................... 203
15.1 Sum of Powers of Numbers .............................. 205
15.2 Other Representations of Sp(n) ........................ 212
15.3 Euler Polynomials and Euler Numbers ................... 217
15.4 Polynomial Expansions Involving Bn(x) ................. 223
Appendix A Newton-Gregory Expansions ......................... 227
Appendix В Generalized Bernoulli and Euler Polynomials ....... 231
B.1 Basic Properties of Bk(α)(x) and Ek(α)(x) ............... 232
B.2 Generalized Bernoulli and Euler Polynomial
Derivative Expansions ................................. 239
B.3 Additional Considerations Involving Newton Series ..... 247
Bibliography .................................................. 253
Index ......................................................... 257
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