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 socalled 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 socalled 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 wellknown
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.