On the Higher-Order Sheffer Orthogonal Polynomial Sequences

On the Higher-Order Sheffer Orthogonal Polynomial Sequences sheds light on the existence/non-existence of B-Type 1 orthogonal polynomials. This book presents a template for analyzing potential orthogonal polynomial sequences including additional higher-order Sheffer classes. This text not only shows that there are no OPS for the special case the B-Type 1 class, but that there are no orthogonal polynomial sequences for the general B-Type 1 class as well. Moreover, it is quite provocative how the seemingly subtle transition from the B-Type 0 class to the B-Type 1 class leads to a drastically more difficult characterization problem. Despite this issue, a procedure is established that yields a definite answer to our current characterization problem, which can also be extended to various other characterization problems as well. Accessible to undergraduate students in the mathematical sciences and related fields, This book functions as an important reference work regarding the Sheffer sequences. The author takes advantage of Mathematica 7 to display unique detailed code and increase the reader's understanding of the implementation of Mathematica 7 and facilitate further experimentation. In addition, this book provides an excellent example of how packages like Mathematica 7 can be used to derive rigorous mathematical results. Table of Contents Cover On the Higher-Order Sheffer Orthogonal Polynomial Sequences ISBN 9781461459682 ISBN 9781461459699 Preface Contents The Sheffer A-Type 0 Orthogonal Polynomial Sequences and Related Results 1.1 Preliminaries 1.2 Sheffer's Analysis of the Type 0 Polynomial Sequences 1.3 The Type 0 Orthogonal Polynomials 1.4 An Overview of the Classification of Type 1.4.1 The Sheffer A-Type Classification 1.4.2 The Sheffer B-Type Classification 1.4.3 The Sheffer C-Type Classification 1.4.4 A Summary of the Rainville s-Type Classifiation 1.5 A Brief Discussion of Meixner's Analysis 1.5.1 Al-Salam's Extension of Meixner's Characterization Some Applications of the Sheffer A-Type 0 Orthogonal Polynomial Sequences 2.1 Preliminaries 2.2 Differential Equations Part I: The "Inverse Method" 2.3 Differential Equations Part II 2.4 Difference Equations 2.5 Gaussian Quadrature 2.6 Problems A Method for Analyzing a Special Case of the Sheffer B-Type 1 Polynomial Sequences 3.1 Preliminaries 3.2 The Formulation of Our Main Problem 3.3 The B-Type 1 Recursion Coeffcients 3.4 Lower-Order B-Type 1 Polynomials Obtained via Generating Function 3.5 Lower-Order B-Type 1 Polynomials Obtained via Three-Term Recurrence Relation 3.6 Managing the Complexity of the Sheffer B-Type 1 Class 3.7 The Solutions of the Simultaneous, Nonlinear, Algebraic System 3.8 On the Verification of Solutions via Computer Algebra 3.9 Conclusion and Future Considerations