Univariate orthonormal polynomials

This page provides mathematical details on sequences of univariate polynomials which are orthonornal with respect to a distribution \mu.

These sequences are used to build multivariate polynomial basis by tensorization in the polynomial chaos expansion (refer to Functional Chaos Expansion and Multivariate Orthonormal basis).

A sequence of univariate orthonormal polynomials is defined by a three-term recurrence formula. Let by P_n the polynomial of degree n. Then the sequence (P_n)_{n \in \Nset} verifies the relation:

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

where for all n, a_n \geq 0 and c_n < 0.

We detail:

  • some well-known sequences of univariate orthonormal polynomials,

  • how to compute any sequence of univariate orthonormal polynomials.

Orthonormal polynomials with respect to usual probability distributions

Below is a table showing some usual sequences of polynomials orthonormal with respect to continuous distribution \mu.

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 distribution are well-known and detailed in the Table below.

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 the degree m-1. Indeed, for the degree n=m, some factors of the denominators of the recurrence coefficients would be equal to zero.

The Table below sums up the available polynomials sequences and the distribution with respect to which they are orthonormal.

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

Orthonormal 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 \mu.

The 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.

For all arbitrary distributions, the three-term recurrence is computed (see AdaptiveStieltjesAlgorithm) and used to build its orthonormal polynomial family (see StandardDistributionPolynomialFactory).