# Orthogonal polynomialsΒΆ

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:

- See the available orthogonal basis.

Examples: