Page 13 - Determinants and Their Applications in Mathematical Physics

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4.12.2 The Generalized Geometric Series and Eulerian
Polynomials ...................... 157
4.12.3 A Further Generalization of the Geometric Series . 162
4.13 Hankelians 6 .......................... 165
4.13.1 Two Matrix Identities and Their Corollaries .... 165
4.13.2 The Factors of a Particular Symmetric Toeplitz
Determinant ...................... 168
4.14 Casoratians — A Brief Note ................. 169

5 Further Determinant Theory 170
5.1 Determinants Which Represent Particular Polynomials . . 170
5.1.1 Appell Polynomial ................... 170
5.1.2 The Generalized Geometric Series and Eulerian
Polynomials ...................... 172
5.1.3 Orthogonal Polynomials ............... 174
5.2 The Generalized Cusick Identities .............. 178
5.2.1 Three Determinants .................. 178
5.2.2 Four Lemmas ...................... 180
5.2.3 Proof of the Principal Theorem ........... 183
5.2.4 Three Further Theorems ............... 184
5.3 The Matsuno Identities .................... 187
5.3.1 A General Identity .................. 187
5.3.2 Particular Identities .................. 189
5.4 The Cofactors of the Matsuno Determinant ........ 192
5.4.1 Introduction ...................... 192
5.4.2 First Cofactors ..................... 193
5.4.3 First and Second Cofactors .............. 194
5.4.4 Third and Fourth Cofactors ............. 195
5.4.5 Three Further Identities ............... 198
5.5 Determinants Associated with a Continued Fraction . . . 201
5.5.1 Continuants and the Recurrence Relation ..... 201
5.5.2 Polynomials and Power Series ............ 203
5.5.3 Further Determinantal Formulas .......... 209
5.6 Distinct Matrices with Nondistinct Determinants ..... 211
5.6.1 Introduction ...................... 211
5.6.2 Determinants with Binomial Elements ....... 212
5.6.3 Determinants with Stirling Elements ........ 217
5.7 The One-Variable Hirota Operator ............. 221
5.7.1 Definition and Taylor Relations ........... 221
5.7.2 A Determinantal Identity ............... 222
5.8 Some Applications of Algebraic Computing ........ 226
5.8.1 Introduction ...................... 226
5.8.2 Hankel Determinants with Hessenberg Elements . 227
5.8.3 Hankel Determinants with Hankel Elements .... 229