15
Orthogonal Polynomials: General Aspects
Abstract
Topics of this chapter are fundamental aspects of orthogonal polynomials.
Orthogonal polynomials play a prominent role for many applications in physics.
In this chapter we discuss in detail Jacobi polynomials and Gegenbauer polynomials in the complex domain and generalizations for non-integer degrees. The
evaluation is either based on recurrence equations or on the hypergeometric
function. The computation of the polynomial roots is based on the eigenvalues
of the corresponding Jacobi matrix. For all evaluations m-codes are available for
download. With respect to computational aspects a superclass is derived with
various methods from creating plots up to integrating weighted scalar products
of orthogonal polynomials. Due to the huge number of orthogonal polynomials
an additional class serving as container for arbitrary orthogonal polynomials is
derived. As an application Zernike polynomials are presented. The corresponding
programming codes are available for download.
Topics of this chapter are some general aspects of orthogonal polynomials. Orthogonal polynomials have many important applications in physics. Therefore some
orthogonal polynomials, e.g., Hermite polynomials, Laguerre polynomials, Legendre polynomials or Chebyshev polynomials, will be discussed in separate chapters.
Orthogonal polynomials treated in this chapter are the Jacobi polynomials and the
Gegenbauer or ultraspherical polynomials, and as an example Zernike polynomials.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_15
181
Orthogonal Polynomials: General Aspects
Abstract
Topics of this chapter are fundamental aspects of orthogonal polynomials.
Orthogonal polynomials play a prominent role for many applications in physics.
In this chapter we discuss in detail Jacobi polynomials and Gegenbauer polynomials in the complex domain and generalizations for non-integer degrees. The
evaluation is either based on recurrence equations or on the hypergeometric
function. The computation of the polynomial roots is based on the eigenvalues
of the corresponding Jacobi matrix. For all evaluations m-codes are available for
download. With respect to computational aspects a superclass is derived with
various methods from creating plots up to integrating weighted scalar products
of orthogonal polynomials. Due to the huge number of orthogonal polynomials
an additional class serving as container for arbitrary orthogonal polynomials is
derived. As an application Zernike polynomials are presented. The corresponding
programming codes are available for download.
Topics of this chapter are some general aspects of orthogonal polynomials. Orthogonal polynomials have many important applications in physics. Therefore some
orthogonal polynomials, e.g., Hermite polynomials, Laguerre polynomials, Legendre polynomials or Chebyshev polynomials, will be discussed in separate chapters.
Orthogonal polynomials treated in this chapter are the Jacobi polynomials and the
Gegenbauer or ultraspherical polynomials, and as an example Zernike polynomials.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_15
181