Chapter 3
Orthogonal Polynomials
The subject of orthogonal polynomials has important applications in physics, quantum mechanics,
mathematical statistics, and other areas in mathematics.
Orthogonal systems of polynomials play an important role in analysis mainly because functions
belonging to general classes can be expanded in series or orthogonal polynomials, such as Fourier
series where the orthogonal system or polynomials is
\[\{\cos nx\}^{\infty }_{n=1}.\]
Emphasis will be placed on Legendre, Chebyshev, Jacobi, Laguerre and Hermite polynomials. We
shall also present some applications.
3.1 Orthogonality
3.2 Legendre Polynomials
3.2.1 Generating Function
3.2.2 Orthogonality of Legendre Polynomials
3.2.3 Expansion of a Function using Legendre
3.3 Jacobi Polynomials
3.4 Chebyshev Polynomials of the First Kind
3.5 Chebyshev Polynomials of the Second Kind
3.6 Hermite Polynomials
3.7 Laguerre Polynomials
3.8 Applications of Orthogonal Polynomials
3.8.1 Applications in Statistics
3.9 Practice Problems
3.2 Legendre Polynomials
3.2.1 Generating Function
3.2.2 Orthogonality of Legendre Polynomials
3.2.3 Expansion of a Function using Legendre
3.3 Jacobi Polynomials
3.4 Chebyshev Polynomials of the First Kind
3.5 Chebyshev Polynomials of the Second Kind
3.6 Hermite Polynomials
3.7 Laguerre Polynomials
3.8 Applications of Orthogonal Polynomials
3.8.1 Applications in Statistics
3.9 Practice Problems