Orthogonal polynomials¶
This page provides mathematical details on sequences of orthogonal polynomials. Some of these sequences will be used to construct the basis of the socalled polynomial chaos expansion.
Mathematical framework¶
Orthogonal polynomials are associated to an inner product, defined as follows. Given an interval of orthogonality (, , ) and a weight function , the 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 is 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 socalled Charlier and Krawtchouk polynomials:
Ortho. poly. 
Probability mass function 
Recurrence coefficients 


Charlier 

Krawtchouk 
Notice that 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 family 
In the library 

Normal 
Hermite 

Uniform 
Legendre 

Gamma 
Laguerre 

Beta 
Jacobi 

Poisson 
Charlier 

Binomial 
Krawtchouk 

Negative Binomial 
Meixner 
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 GramSchmidt 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.