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 so-called polynomial chaos expansion.

Mathematical framework¶

Orthogonal polynomials are associated to an inner product, defined as follows. Given an interval of orthogonality $[\alpha,\beta]$ ( $\alpha \in \Rset \cup \{-\infty\}$ , $\beta \in \Rset \cup \{\infty\}$ , $\alpha < \beta$ ) and a weight function $w(x)> 0$ , the polynomials $P$ and $Q$ are orthogonal if:

$\scalarproduct{P}{Q} = \int_{\alpha}^{\beta}P(x)Q(x)~w(x) dx = 0$

Therefore, a sequence of orthogonal polynomials $(P_n)_{n\geq 0}$ ( $P_n$ : polynomial of degree $n$ ) verifies:

$\scalarproduct{P_m}{P_n} = 0 \text{~~for every~~} m \neq n$

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

$\parallel P_n\parallel = \scalarproduct{P_n}{P_n}^{1/2}$

In the following, we consider weight functions $w(x)$ corresponding to probability density functions, which satisfy:

$\int_{\alpha}^{\beta} \; w(x) \; dx \, \, = \,\, 1$

Moreover, we consider families of orthonormal polynomials $(P_n)_{n\geq 0}$ , that is polynomials with a unit norm:

$\|P_n\| \, \, = \, \, 1$

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

$P_{n+1}\ =\ (a_nx+b_n)\ P_n\ +\ c_n\ P_{n-1}$

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.	$P_n(x)$	Weight $w(x)^{\strut}$	Recurrence coefficients $(a_n,b_n,c_n)$
Hermite	${He}_n(x)^{\strut}$	$\displaystyle \frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}}$	$\begin{array}{ccc} a_n & = & \frac{1}{\sqrt{n+1}} \\ b_n & = & 0 \\ c_n & = & - \sqrt{\frac{n}{n+1}} \end{array}$
Legendre	$\begin{array}{c} {Le}_n(x) \\ \\ \alpha>-1 \\ \end{array}$	$\displaystyle \frac{1}{2}^{\strut} \times \mathbb{I}_{[-1,1]}(x)$	$\begin{array}{ccc} a_n & = & \frac{\sqrt{(2n+1)(2n+3)}}{n+1} \\ b_n & = & 0 \\ c_n & = & -\frac{ n \sqrt{2n+3} }{ (n+1)\sqrt{2n-1} } \end{array}$
Laguerre	$L_n^{(\alpha)}(x)$	$\displaystyle \frac{x^{k-1}}{\Gamma(k)}~e^{-x} \mathbb{I}_{[0,+\infty[}(x)$	$\begin{array}{ccc} \omega_{n} & = & \left((n+1)(n+k+1) \right)^{-1/2} \\ a_n & = & \omega_{n} \\ b_n & = & -(2n+k+1)~\omega_{n} \\ c_n & = & -\sqrt{(n+k)n}~\omega_{n} \end{array}$
Jacobi	$\begin{array}{c} J^{(\alpha,\beta)}_n(x) \\ \\ \\ \alpha,\beta>-1 \\ \end{array}$ \| $\frac{(1-x)^{\alpha}(1+x)^{\beta}}{2^{\alpha + \beta + 1} B(\beta + 1, \alpha + 1)} \mathbb{I}_{[-1,1]}(x)$		$\begin{array}{ccc} K_{1,n} & = & \frac{2n+\alpha + \beta + 3}{(n+1)(n+\alpha+1)(n+\beta+1)(n+\alpha+\beta+1)} \\ \\ K_{2,n} & = & \frac{1}{2} \sqrt{(2n + \alpha + \beta + 1) K_{1,n}} \\ \\a_n & = & K_{2,n}(2n+\alpha + \beta + 2) \\ \\ b_n & = & K_{2,n}\frac{(\alpha - \beta)(\alpha + \beta)}{2n+\alpha+\beta} \\ \\ c_n & = & - \frac{2n+\alpha+\beta + 2}{2n+\alpha+\beta} \Big[(n+\alpha)(n+\beta) \\ & & \times (n+\alpha+\beta)n\frac{K_{1,n}}{2n+\alpha+\beta-1}\Big]^{1/2} \end{array}$

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

Ortho. poly.	$P_n(x)$	Probability mass function	Recurrence coefficients $(a_n,b_n,c_n)$
Charlier	$\begin{array}{c} Ch^{(\lambda)}_n(x) \\ \\ \lambda>0 \\ \end{array}$	$\begin{array}{c} \displaystyle{\frac{\lambda^k}{k!}~e^{-\lambda}} \\ \\ k=0,1,2,\dots \\ \end{array}$	$\begin{array}{ccc} a_n & = & - \frac{1}{\sqrt{\lambda (n+1)}} \\ \\ b_n & = & \frac{n+\lambda}{\sqrt{\lambda (n+1)}} \\ \\ c_n & = & - \sqrt{1 - \frac{1}{n+1}} \end{array}$
Krawtchouk $^{\dagger}$	$\begin{array}{c} Kr^{(m,p)}_n(x) \\ \\ m \in \Nset~,~p \in [0,1] \\ \end{array}$	$\begin{array}{c} \displaystyle{\binom{m}{k}p^k (1-p)^{m-k}} \\ \\ k=0,1,2,\dots \\ \end{array}$	$\begin{array}{ccc} a_n & = & - \frac{1}{\sqrt{(n+1)(m-n)p(1-p)}} \\ \\ b_n & = & \frac{p(m-n)+n(1-p)}{\sqrt{(n+1)(m-n)p(1-p)}} \\ \\ c_n & = & - \sqrt{(1 - \frac{1}{n+1})(1+\frac{1}{m-n})} \end{array}$

Notice that the Krawtchouk polynomials are only defined up to a degree $n$ equal to $m-1$ . Indeed, for $n=m$ , 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 $\cN(0,1)$	$\Rset$	Hermite	`HermiteFactory`
Uniform $\cU(-1,1)$	$[-1,1]$	Legendre	`LegendreFactory`
Gamma $\Gamma(k,1,0)$	$(0,+\infty)$	Laguerre	`LaguerreFactory`
Beta $B(\alpha,\beta,-1,1)$	$(-1,1)$	Jacobi	`JacobiFactory`
Poisson $\cP(\lambda)$	$\Nset$	Charlier	`CharlierFactory`
Binomial $\cB(m,p)$	$\{0,\dots,m\}$	Krawtchouk $^{\dagger}$	`KrawtchoukFactory`
Negative Binomial $\cN \cB(m,p)$	$\Nset$	Meixner	`MeixnerFactory`