Page 12 - Determinants and Their Applications in Mathematical Physics

Contents xi

4.4.3 The Generalized Hyperbolic Functions ....... 81
4.5 Centrosymmetric Determinants ............... 85
4.5.1 Definition and Factorization ............. 85
4.5.2 Symmetric Toeplitz Determinants .......... 87
4.5.3 Skew-Centrosymmetric Determinants ........ 90
4.6 Hessenbergians ......................... 90
4.6.1 Definition and Recurrence Relation ......... 90
4.6.2 A Reciprocal Power Series .............. 92
4.6.3 A Hessenberg–Appell Characteristic Polynomial . 94
4.7 Wronskians ........................... 97
4.7.1 Introduction ...................... 97
4.7.2 The Derivatives of a Wronskian ........... 99
4.7.3 The Derivative of a Cofactor ............. 100
4.7.4 An Arbitrary Determinant .............. 102
4.7.5 Adjunct Functions ................... 102
4.7.6 Two-Way Wronskians ................. 103
4.8 Hankelians 1 .......................... 104
4.8.1 Definition and the φ m Notation ........... 104
4.8.2 Hankelians Whose Elements are Differences .... 106
4.8.3 Two Kinds of Homogeneity .............. 108
4.8.4 The Sum Formula ................... 108
4.8.5 Turanians ........................ 109
4.8.6 Partial Derivatives with Respect to φ m ....... 111
4.8.7 Double-Sum Relations ................ 112
4.9 Hankelians 2 .......................... 115
4.9.1 The Derivatives of Hankelians with Appell
Elements ........................ 115
4.9.2 The Derivatives of Turanians with Appell and
Other Elements .................... 119
4.9.3 Determinants with Simple Derivatives of All
Orders .......................... 122
4.10 Henkelians 3 .......................... 123
4.10.1 The Generalized Hilbert Determinant ........ 123
4.10.2 Three Formulas of the Rodrigues Type ....... 127
4.10.3 Bordered Yamazaki–Hori Determinants — 1 .... 129
4.10.4 A Particular Case of the Yamazaki–Hori
Determinant ...................... 135
4.11 Hankelians 4 .......................... 137
4.11.1 v-Numbers ....................... 137
4.11.2 Some Determinants with Determinantal Factors . 138
4.11.3 Some Determinants with Binomial and Factorial
Elements ........................ 142
4.11.4 A Nonlinear Differential Equation .......... 147
4.12 Hankelians 5 .......................... 153
4.12.1 Orthogonal Polynomials ............... 153