This book defines sets of orthogonal polynomials and derives a number of properties satisfied by any such set. It continues by describing the classical orthogonal polynomials and the additional properties they have.
The first chapter defines the orthogonality condition for two functions. It then gives an iterative process to produce a set of polynomials which are orthogonal to one another and then describes a number of properties satisfied by any set of orthogonal polynomials. The classical orthogonal polynomials arise when the weight function in the orthogonality condition has a particular form. These polynomials have a further set of properties and in particular satisfy a second order differential equation.
Each subsequent chapter investigates the properties of a particular polynomial set starting from its differential equation.
- Definitions and General Properties
- Hermite Polynomials
- Associated Laguerre Polynomials
- Legendre Polynomials
- Chebyshev Polynomials of the First Kind
- Chebyshev Polynomials of the Second Kind
- Chebyshev Polynomials of the Third Kind
- Chebyshev Polynomials of the Fourth Kind
- Gegenbauer Polynomials
- Associated Legendre Functions
- Jacobi Polynomials
- General Appendix
Readership: Undergraduate and graduate students.