Orthogonal Polynomials

Orthogonal polynomials are a sequence of polynomials that are orthogonal with respect to a specific inner product. They are widely used in numerical analysis, approximation theory, and other fields. Here we list some important orthogonal polynomials and their properties.

Orthogonal Polynomial	Definition on
Jacobi
Jacobi (shifted)
Gegenbauer or Ultraspherical
Legendre
Chebyshev (first kind)
Chebyshev (first kind, shifted)
Chebyshev (second kind)
Chebyshev (second kind, shifted)
Laguerre
Hermite ( or )

where is the Pochhammer symbol, satisfying

provided that is a nonnegative integer. The Pochhammer symbol is related to the gamma function by the formula

And is the hypergeometric function, defined as

The series terminates if or is a nonpositive integer, in which case the function reduces to a polynomial:

Jacobi Polynomials

Hypergeometric Definition
The Jacobi polynomials are defined via the hypergeometric function as follows:

In this case, the series for the hypergeometric function is finite, therefore one obtain the following equivalent expression:

Differential Definition
Legendre polynomials are a special case of Jacobi polynomials with , having the following differential definition:

In fact, the general Jacobi polynomials also have similar differential expression:

which is called the Rodrigues’ formula.

Properties

Orthogonality
The Jacobi polynomials are orthogonal with respect to the weight function on the interval , i.e.,

where

Hence the Jacobi polynomials are not orthonormal in general, but can be normalized by dividing by the square root of the corresponding , and an alternative normalization is sometimes preferred due to its simplicity:

Symmetry Relation
Jacobi polynomials satisfy the symmetry relation

thus the other terminal value is

Derivatives
The th derivative of the explict expression of Jacobi polynomials leads to

therefore the Jacobi-based th differential operator is

(first elements of first row are ), which satisfies

The differential operator, as well as multiplication and conversion operators are introduced in [6] for the ultraspherical spectral method.

Differential Equation
The Jacobi polynomial is a solution to the following 2nd-order linear homogeneous differential equation:

hence is a eigenfunction of the corresponding Sturm-Liouville operator.

Recurrence Relation
As an orthogonal polynomial, Jacobi polynomials satisfy the following three-term recurrence relation:

where

The multiplication operator , which satisfies

is given by

Generally, we need the operator for solving a variable coefficient differential equation by spectral methods in spectral space, which is assumed to satisfy

Since no closed-form expression is known, can only be constructed via the recurrence relation

where

Linearity of leads to

and the recurrence coefficients can be calculated by as follows:

Now we can construct the operator provided that is represented by a Jacobi series.

On the other hand, like the ultraspherical case, we need the conversion operator to improve the order of polynomials used to represent the solution. The conversion operator is defined as

To construct , differentiate both sides of results

recall the derivative of Jacobi polynomials satisfies

therefore we have

Notice can be represented by , and again:

we have

where

Therefore, the conversion operator is

Now the linear differential operator

can be approximated by Jacobi-based operator

which maps a series to a series.

Generating Function
The generating function of Jacobi polynomials is given by

where

and the branch of square root is chosen s.t. .

Reference

1. Orthogonal Polynomials
2. Hypergeometric Function
3. Jacobi Polynomials
4. Yudell L. Luke, The Special Functions and Their Approximations, Volume 1, Academic Press, 1969.
5. Mason, J. C., Handscomb, D. C., Chebyshev Polynomials, Chapman and Hall/CRC, 2003.
6. Sheehan Olver, Alex Townsend. A Fast and Well-Conditioned Spectral Method, SIAM Rev. 55, 2013