# Orthogonal polynomials¶

This section provides some mathematical details on sequences of orthogonal polynomials. Some of these sequences will be used to construct the basis of the so-called polynomial chaos expansion.
Mathematical framework
The orthogonal polynomials are associated to an inner product, defined as follows:
Given an interval of orthogonality (, , ) and a weight function , every pair of polynomials and are orthogonal if:

Therefore, a sequence of orthogonal polynomials (: polynomial of degree ) verifies:

The chosen inner product induces a norm on polynomials in the usual way:

In the following, we consider weight functions corresponding to probability density functions, which satisfy:

Moreover, we consider families of orthonormal polynomials , that is polynomials with a unit norm:

Any sequence of orthogonal polynomials has a recurrence formula relating any three consecutive polynomials as follows:

Orthogonormal polynomials with respect to usual probability distributions
Below, a table showing an example of particular (normalized) orthogonal polynomials associated with continuous weight functions.
Note that the orthonormal polynomials are orthonormal with respect to the standard representative distribution of the given distribution.
Ortho. poly. Weight Recurrence coefficients
Hermite
Legendre
Laguerre
Jacobi |

Furthermore, two families of orthonormal polynomials with respect to discrete probability distributions are available, namely the so-called Charlier and Krawtchouk polynomials:

Ortho. poly. Probability mass function Recurrence coefficients
Charlier
Krawtchouk
The Krawtchouk polynomials are only defined up to a degree equal to . Indeed, for , some factors of the denominators of the recurrence coefficients would be equal to zero.
To sum up, the distribution type are reported in the table below together with the associated families of orthonormal polynomials.
Distribution Support Polynomial
Normal Hermite
Uniform Legendre
Gamma Laguerre
Beta Jacobi
Poisson Charlier
Binomial Krawtchouk
It is recalled that the Krawtchouk polynomials are only defined up to a degree equal to .
Orthogonal polynomials with respect to arbitrary probability distributions
It is also possible to generate a family of orthonormal polynomials with respect to an arbitrary probability distribution . The well-known Gram-Schmidt algorithm can be used to this end. Note that this algorithm gives a constructive proof of the existence of orthonormal bases.
However it is known to be numerically unstable, so alternative procedures are often used in practice. The available orthonormalization algorithm is the Stieltjes algorithm.

API:

Examples: