Classical orthogonal polynomials

Classical orthogonal polynomials

In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the Legendre polynomials.[1]

They have many important applications in such areas as mathematical physics (in particular, the theory of random matrices), approximation theory, numerical analysis, and many others.

Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien-Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, the study of continued fractions by P. L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials.

The classical orthogonal polynomials are characterized by being solutions to a differential equation of the form

 Q(x) \, f'' +  L(x)\,f' + \lambda  f = 0~.

There are several more general definitions of orthogonal classical polynomials; for example, Andrews & Askey (1985) use the term for all polynomials in the Askey scheme.

Contents

Definition

In general, the orthogonal polynomials Pn with respect to a weight W:RR+ on the real line are defined by

\begin{align}
&\deg P_n = n~, \quad n = 0,1,2,\ldots\\
&\int P_m(x) \, P_n(x) \, W(x)\,dx = 0~, \quad m \neq n~.
\end{align}

The relations above define Pn up to multiplication by a number. Various normalisations are used to fix the constant, e.g.

 \int P_n(x)^2 W(x)\,dx = 1~.

The classical orthogonal polynomials correspond to the three families of weights:

\begin{align}
\text{(Jacobi)}\quad &W(x) = \begin{cases} 
  (1 - x)^\alpha (1+x)^\beta~, & -1 \leq x \leq 1 \\
  0~, &\text{otherwise}
\end{cases}  \\
\text{(Hermite)}\quad  &W(x) = \exp(- x^2) \\
\text{(Laguerre)}\quad &W(x) = \begin{cases}
  x^\alpha \exp(- x)~, &\quad x \geq 0 \\
  0~, &\text{otherwise}
\end{cases}      
\end{align}

The standard normalisation (also called standartisation) is detailed below.

Jacobi polynomials

For α,β > −1 the Jacobi polynomials are given by the formula

P_n^{(\alpha,\beta)} (z)
= \frac{(-1)^n}{2^n n!} (1-z)^{-\alpha} (1+z)^{-\beta}
\frac{d^n}{dz^n} \left\{ (1-z)^\alpha (1+z)^\beta (1 - z^2)^n \right\}~.

They are normalised (standartised) by

P_n^{(\alpha, \beta)} (1) = {n+\alpha\choose n},

and satisfy the orthogonality condition

\begin{align}
&\int_{-1}^1 (1-x)^{\alpha} (1+x)^{\beta} 
P_m^{(\alpha,\beta)} (x)P_n^{(\alpha,\beta)} (x) \; dx \\
&\quad=
\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}
\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!} \delta_{nm}.
\end{align}

The Jacobi polynomials are solutions to the differential equation


(1-x^2)y'' + ( \beta-\alpha - (\alpha + \beta + 2)x )y'+ n(n+\alpha+\beta+1) y = 0~.

Important special cases

The Jacobi polynomials with α = β are called the Gegenbauer polynomials (with parameter γ = α + ½).

For α = β = 0, these are called the Legendre polynomials (for which the interval of orthogonality is [−1, 1] and the weight function is simply 1):


P_0(x) = 1,\, P_1(x) = x,\,P_2(x) = \frac{3x^2-1}{2},\,
P_3(x) = \frac{5x^3-3x}{2},\ldots

For α = β = ±½, one obtains the Chebyshev polynomials (of the second and first kind, respectively).

Hermite polynomials

The Hermite polynomials are defined by[2]

 H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}=e^{x^2/2}\bigg (x-\frac{d}{dx} \bigg )^n e^{-x^2/2}\,\!

They satisfy the orthogonality condition

 \int_{-\infty}^\infty H_n(x) H_m(x) e^{-x^2} dx = \sqrt{\pi} 2^n n! \delta_{mn}~,

and the differential equation

y'' - 2xy' + 2n\,y = 0~.

Laguerre polynomials

The generalised Laguerre polynomials are defined by

L_n^{(\alpha)}(x)=
{x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right)

(the classical Laguerre polynomials correspond to α=0.)

They satisfy the orthogonality relation

\int_0^{\infty}x^\alpha e^{-x} L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)dx=\frac{\Gamma(n+\alpha+1)}{n!}\delta_{n,m}~,

and the differential equation


x\,y'' + (\alpha +1 - x)\,y' + n\,y = 0~.

Differential equation

The classical orthogonal polynomials arise from a differential equation of the form

 Q(x) \, f'' +  L(x)\,f' + \lambda  f = 0 \,

where Q is a given quadratic (at most) polynomial, and L is a given linear polynomial. The function f, and the constant λ, are to be found.

(Note that it makes sense for such an equation to have a polynomial solution.
Each term in the equation is a polynomial, and the degrees are consistent.)

This is a Sturm-Liouville type of equation. Such equations generally have singularities in their solution functions f except for particular values of λ. They can be thought of an eigenvector/eigenvalue problems: Letting D be the differential operator, D(f) = Q f'' + L f'\,, and changing the sign of λ, the problem is to find the eigenvectors (eigenfunctions) f, and the corresponding eigenvalues λ, such that f does not have singularities and D(f) = λf.

The solutions of this differential equation have singularities unless λ takes on specific values. There is a series of numbers \lambda_0, \lambda_1, \lambda_2, \dots\, that lead to a series of polynomial solutions P_0, P_1, P_2, \dots\, if one of the following sets of conditions are met:

  1. Q is actually quadratic, L is linear, Q has two distinct real roots, the root of L lies strictly between the roots of Q, and the leading terms of Q and L have the same sign.
  2. Q is not actually quadratic, but is linear, L is linear, the roots of Q and L are different, and the leading terms of Q and L have the same sign if the root of L is less than the root of Q, or vice-versa.
  3. Q is just a nonzero constant, L is linear, and the leading term of L has the opposite sign of Q.

These three cases lead to the Jacobi-like, Laguerre-like, and Hermite-like polynomials, respectively.

In each of these three cases, we have the following:

  • The solutions are a series of polynomials P_0, P_1, P_2, \dots\,, each P_n\, having degree n, and corresponding to a number {\lambda}_n\,.
  • The interval of orthogonality is bounded by whatever roots Q has.
  • The root of L is inside the interval of orthogonality.
  • Letting R(x) = e^{\int \frac{L(x)}{Q(x)}\,dx}\,, the polynomials are orthogonal under the weight function W(x) =\frac{R(x)}{Q(x)}\,
  • W(x) has no zeros or infinities inside the interval, though it may have zeros or infinities at the end points.
  • W(x) gives a finite inner product to any polynomials.
  • W(x) can be made to be greater than 0 in the interval. (Negate the entire differential equation if necessary so that Q(x) > 0 inside the interval.)

Because of the constant of integration, the quantity R(x) is determined only up to an arbitrary positive multiplicative constant. It will be used only in homogeneous differential equations (where this doesn't matter) and in the definition of the weight function (which can also be indeterminate.) The tables below will give the "official" values of R(x) and W(x).

Rodrigues' formula

Under the assumptions of the preceding section, Pn(x) is proportional to \frac{1}{W(x)} \  \frac{d^n}{dx^n}\left(W(x)[Q(x)]^n\right).

This is known as Rodrigues' formula, after Olinde Rodrigues. It is often written

P_n(x) = \frac{1}{{e_n}W(x)} \  \frac{d^n}{dx^n}\left(W(x)[Q(x)]^n\right)

where the numbers en depend on the standardization. The standard values of en will be given in the tables below.

The numbers λn

Under the assumptions of the preceding section, we have

\lambda_n = - n \left( \frac{n-1}{2} Q'' + L' \right).

(Since Q is quadratic and L is linear, Q'' and L' are constants, so these are just numbers.)

Second form for the differential equation

Let

R(x) = e^{\int \frac{L(x)}{Q(x)}\,dx}\,.

Then

(Ry')' = R\,y'' + R'\,y' = R\,y'' + \frac{R\,L}{Q}\,y'.

Now multiply the differential equation

Q\,y'' + L\,y' + \lambda y = 0\,

by R/Q, getting

R\,y'' + \frac{R\,L}{Q}\,y' + \frac{R\,\lambda}{Q}\,y = 0\,

or

(Ry')' + \frac{R\,\lambda}{Q}\,y = 0.\,

This is the standard Sturm-Liouville form for the equation.

Third form for the differential equation

Let S(x) = \sqrt{R(x)} = e^{\int \frac{L(x)}{2\,Q(x)}\,dx}.\,

Then

S' = \frac{S\,L}{2\,Q}.

Now multiply the differential equation

Q\,y'' + {L}\,y' + \lambda y = 0\,

by S/Q, getting

S\,y'' + \frac{S\,L}{Q}\,y' + \frac{S\,\lambda}{Q}\,y = 0\,

or

S\,y'' + 2\,S'\,y' + \frac{S\,\lambda}{Q}\,y = 0\,

But (S\,y)'' = S\,y'' + 2\,S'\,y' + S''\,y, so

(S\,y)'' + \left(\frac{S\,\lambda}{Q} - S''\right)\,y = 0,\,

or, letting u = Sy,

u'' + \left(\frac{\lambda}{Q} - \frac{S''}{S}\right)\,u = 0.\,

Formulas involving derivatives

Under the assumptions of the preceding section, let P_n^{[r]} denote the rth derivative of Pn. (We put the "r" in brackets to avoid confusion with an exponent.) P_n^{[r]} is a polynomial of degree n − r. Then we have the following:

  • (orthogonality) For fixed r, the polynomial sequence P_r^{[r]}, P_{r+1}^{[r]}, P_{r+2}^{[r]}, \dots are orthogonal, weighted by WQ^r\,.
  • (generalized Rodrigues' formula) P_n^{[r]} is proportional to \frac{1}{W(x)[Q(x)]^r} \  \frac{d^{n-r}}{dx^{n-r}}\left(W(x)[Q(x)]^n\right).
  • (differential equation) P_n^{[r]} is a solution of {Q}\,y'' + (rQ'+L)\,y' + [{\lambda}_n-{\lambda}_r]\,y = 0\,, where {\lambda}_r\, is the same function as {\lambda}_n\,, that is, {\lambda}_r = - r \left( \frac{r-1}{2} Q'' + L' \right)
  • (differential equation, second form) P_n^{[r]} is a solution of (RQ^{r}y')' + [{\lambda}_n-{\lambda}_r]RQ^{r-1}\,y = 0\,

There are also some mixed recurrences. In each of these, the numbers a, b, and c depend on n and r, and are unrelated in the various formulas.

  • P_n^{[r]} = aP_{n+1}^{[r+1]} + bP_n^{[r+1]} + cP_{n-1}^{[r+1]}
  • P_n^{[r]} = (ax+b)P_n^{[r+1]} + cP_{n-1}^{[r+1]}
  • QP_n^{[r+1]} = (ax+b)P_n^{[r]} + cP_{n-1}^{[r]}

There are an enormous number of other formulas involving orthogonal polynomials in various ways. Here is a tiny sample of them, relating to the Chebyshev, associated Laguerre, and Hermite polynomials:

  • 2\,T_{m}(x)\,T_{n}(x) = T_{m+n}(x) + T_{m-n}(x)\,
  • H_{2n}(x) = (-4)^{n}\,n!\,L_{n}^{(-1/2)}(x^2)
  • H_{2n+1}(x) = 2(-4)^{n}\,n!\,x\,L_{n}^{(1/2)}(x^2)

Orthogonality

The differential equation for a particular λ may be written (omitting explicit dependence on x)

Q\ddot{f}_n+L\dot{f}_n+\lambda_nf_n=0

multiplying by (R / Q)fm yields

Rf_m\ddot{f}_n+\frac{R}{Q}Lf_m\dot{f}_n+\frac{R}{Q}\lambda_nf_mf_n=0

and reversing the subscripts yields

Rf_n\ddot{f}_m+\frac{R}{Q}Lf_n\dot{f}_m+\frac{R}{Q}\lambda_mf_nf_m=0

subtracting and integrating:


\int_a^b \left[R(f_m\ddot{f}_n-f_n\ddot{f}_m)+
\frac{R}{Q}L(f_m\dot{f}_n-f_n\dot{f}_m)\right] \, dx
+(\lambda_n-\lambda_m)\int_a^b \frac{R}{Q}f_mf_n \, dx = 0

but it can be seen that


\frac{d}{dx}\left[R(f_m\dot{f}_n-f_n\dot{f}_m)\right]=
R(f_m\ddot{f}_n-f_n\ddot{f}_m)\,\,+\,\,R\frac{L}{Q}(f_m\dot{f}_n-f_n\dot{f}_m)

so that:

\left[R(f_m\dot{f}_n-f_n\dot{f}_m)\right]_a^b\,\,+\,\,(\lambda_n-\lambda_m)\int_a^b \frac{R}{Q}f_mf_n \, dx=0

If the polynomials f are such that the term on the left is zero, and \lambda_m \ne \lambda_n for m \ne n, then the orthogonality relationship will hold:

\int_a^b \frac{R}{Q}f_mf_n \, dx=0

for m \ne n.

Derivation from differential equation

All of the polynomial sequences arising from the differential equation above are equivalent, under scaling and/or shifting of the domain, and standardizing of the polynomials, to more restricted classes. Those restricted classes are exactly "classical orthogonal polynomials".

  • Every Jacobi-like polynomial sequence can have its domain shifted and/or scaled so that its interval of orthogonality is [−1, 1], and has Q = 1 − x2. They can then be standardized into the Jacobi polynomials P_n^{(\alpha, \beta)}. There are several important subclasses of these: Gegenbauer, Legendre, and two types of Chebyshev.
  • Every Laguerre-like polynomial sequence can have its domain shifted, scaled, and/or reflected so that its interval of orthogonality is [0, \infty), and has Q = x. They can then be standardized into the Associated Laguerre polynomials L_n^{(\alpha)}. The plain Laguerre polynomials \ L_n are a subclass of these.
  • Every Hermite-like polynomial sequence can have its domain shifted and/or scaled so that its interval of orthogonality is (-\infty, \infty), and has Q = 1 and L(0) = 0. They can then be standardized into the Hermite polynomials H_n\,.

Because all polynomial sequences arising from a differential equation in the manner described above are trivially equivalent to the classical polynomials, the actual classical polynomials are always used.

Jacobi polynomial

The Jacobi-like polynomials, once they have had their domain shifted and scaled so that the interval of orthogonality is [−1, 1], still have two parameters to be determined. They are α and β in the Jacobi polynomials, written P_n^{(\alpha, \beta)}. We have Q(x) = 1-x^2\, and L(x) = \beta-\alpha-(\alpha+\beta+2)\, x. Both α and β are required to be greater than −1. (This puts the root of L inside the interval of orthogonality.)

When α and β are not equal, these polynomials are not symmetrical about x = 0.

The differential equation

(1-x^2)\,y'' + (\beta-\alpha-[\alpha+\beta+2]\,x)\,y' + {\lambda}\,y = 0\qquad \mathrm{with}\qquad\lambda = n(n+1+\alpha+\beta)\,

is Jacobi's equation.

For further details, see Jacobi polynomials.

Gegenbauer polynomials

When one sets the parameters α and β in the Jacobi polynomials equal to each other, one obtains the Gegenbauer or ultraspherical polynomials. They are written C_n^{(\alpha)}, and defined as

C_n^{(\alpha)}(x) = \frac{\Gamma(2\alpha\!+\!n)\,\Gamma(\alpha\!+\!1/2)}
{\Gamma(2\alpha)\,\Gamma(\alpha\!+\!n\!+\!1/2)}\! \  P_n^{(\alpha-1/2, \alpha-1/2)}.

We have Q(x) = 1-x^2\, and L(x) = -(2\alpha+1)\, x. \alpha\, is required to be greater than −1/2.

(Incidentally, the standardization given in the table below would make no sense for α = 0 and n ≠ 0, because it would set the polynomials to zero. In that case, the accepted standardization sets C_n^{(0)}(1) = \frac{2}{n} instead of the value given in the table.)

Ignoring the above considerations, the parameter α is closely related to the derivatives of C_n^{(\alpha)}:

C_n^{(\alpha+1)}(x) = \frac{1}{2\alpha}\! \  \frac{d}{dx}C_{n+1}^{(\alpha)}(x)

or, more generally:

C_n^{(\alpha+m)}(x) = \frac{\Gamma(\alpha)}{2^m\Gamma(\alpha+m)}\! \  C_{n+m}^{(\alpha)[m]}(x).

All the other classical Jacobi-like polynomials (Legendre, etc.) are special cases of the Gegenbauer polynomials, obtained by choosing a value of α and choosing a standardization.

For further details, see Gegenbauer polynomials.

Legendre polynomials

The differential equation is

(1-x^2)\,y'' - 2x\,y' + {\lambda}\,y = 0\qquad \mathrm{with}\qquad\lambda = n(n+1).\,

This is Legendre's equation.

The second form of the differential equation is:

([1-x^2]\,y')' + \lambda\,y = 0.\,

The recurrence relation is

(n+1)\,P_{n+1}(x) = (2n+1)x\,P_n(x) - n\,P_{n-1}(x).\,

A mixed recurrence is

P_{n+1}^{[r+1]}(x) = P_{n-1}^{[r+1]}(x) + (2n+1)\,P_n^{[r]}(x).\,

Rodrigues' formula is

P_n(x) = \,\frac{n!}{2n!} \  \frac{d^n}{dx^n}\left([x^2-1]^n\right).

For further details, see Legendre polynomials.

Associated Legendre polynomials

The Associated Legendre polynomials, denoted P_\ell^{(m)}(x) where \ell and m are integers with 0 \leqslant m  \leqslant \ell, are defined as

P_\ell^{(m)}(x) = (-1)^m\,(1-x^2)^{m/2}\ P_\ell^{[m]}(x).\,

The m in parentheses (to avoid confusion with an exponent) is a parameter. The m in brackets denotes the mth derivative of the Legendre polynomial.

These "polynomials" are misnamed—they are not polynomials when m is odd.

They have a recurrence relation:

(\ell+1-m)\,P_{\ell+1}^{(m)}(x) = (2\ell+1)x\,P_\ell^{(m)}(x) - (\ell+m)\,P_{\ell-1}^{(m)}(x).\,

For fixed m, the sequence P_m^{(m)}, P_{m+1}^{(m)}, P_{m+2}^{(m)}, \dots are orthogonal over [−1, 1], with weight 1.

For given m, P_\ell^{(m)}(x) are the solutions of

(1-x^2)\,y'' -2xy' + \left[\lambda - \frac{m^2}{1-x^2}\right]\,y = 0\qquad \mathrm{with}\qquad\lambda = \ell(\ell+1).\,

Chebyshev polynomials

The differential equation is

(1-x^2)\,y'' - x\,y' + {\lambda}\,y = 0\qquad \mathrm{with}\qquad\lambda = n^2.\,

This is Chebyshev's equation.

The recurrence relation is

T_{n+1}(x) = 2x\,T_n(x) - T_{n-1}(x).\,

Rodrigues' formula is

T_n(x) = \frac{\Gamma(1/2)\sqrt{1-x^2}}{(-2)^n\,\Gamma(n+1/2)} \  \frac{d^n}{dx^n}\left([1-x^2]^{n-1/2}\right).

These polynomials have the property that, in the interval of orthogonality,

T_n(x) = \cos(n\,\arccos(x)).

(To prove it, use the recurrence formula.)

This means that all their local minima and maxima have values of −1 and +1, that is, the polynomials are "level". Because of this, expansion of functions in terms of Chebyshev polynomials is sometimes used for polynomial approximations in computer math libraries.

Some authors use versions of these polynomials that have been shifted so that the interval of orthogonality is [0, 1] or [−2, 2].

There are also Chebyshev polynomials of the second kind, denoted U_n\,

We have:

U_n = \frac{1}{n+1}\,T_{n+1}'.\,

For further details, including the expressions for the first few polynomials, see Chebyshev polynomials.

Laguerre polynomials

The most general Laguerre-like polynomials, after the domain has been shifted and scaled, are the Associated Laguerre polynomials (also called Generalized Laguerre polynomials), denoted L_n^{(\alpha)}. There is a parameter α, which can be any real number strictly greater than −1. The parameter is put in parentheses to avoid confusion with an exponent. The plain Laguerre polynomials are simply the α = 0 version of these:

L_n(x) = L_n^{(0)}(x).\,

The differential equation is

x\,y'' + (\alpha + 1-x)\,y' + {\lambda}\,y = 0\text{ with }\lambda = n.\,

This is Laguerre's equation.

The second form of the differential equation is

(x^{\alpha+1}\,e^{-x}\, y')' + {\lambda}\,x^\alpha \,e^{-x}\,y = 0.\,

The recurrence relation is

(n+1)\,L_{n+1}^{(\alpha)}(x) = (2n+1+\alpha-x)\,L_n^{(\alpha)}(x) - (n+\alpha)\,L_{n-1}^{(\alpha)}(x).\,

Rodrigues' formula is

L_n^{(\alpha)}(x) = \frac{x^{-\alpha}e^x}{n!} \  \frac{d^n}{dx^n}\left(x^{n+\alpha}\,e^{-x}\right).

The parameter α is closely related to the derivatives of L_n^{(\alpha)}:

L_n^{(\alpha+1)}(x) = - \frac{d}{dx}L_{n+1}^{(\alpha)}(x)

or, more generally:

L_n^{(\alpha+m)}(x) = (-1)^m L_{n+m}^{(\alpha)[m]}(x).

Laguerre's equation can be manipulated into a form that is more useful in applications:

u = x^{\frac{\alpha-1}{2}}e^{-x/2}L_n^{(\alpha)}(x)

is a solution of

u'' + \frac{2}{x}\,u' + \left[\frac{\lambda}{x} - \frac{1}{4} - \frac{\alpha^2-1}{4x^2}\right]\,u = 0\text{ with } \lambda = n+\frac{\alpha+1}{2}.\,

This can be further manipulated. When \ell = \frac{\alpha-1}{2} is an integer, and n \ge \ell+1:

u = x^\ell e^{-x/2} L_{n-\ell-1}^{(2\ell+1)}(x)

is a solution of

u'' + \frac{2}{x}\,u' + \left[\frac{\lambda}{x} - \frac{1}{4} - \frac{\ell(\ell+1)}{x^2}\right]\,u = 0\text{ with }\lambda = n.\,

The solution is often expressed in terms of derivatives instead of associated Laguerre polynomials:

u = x^{\ell}e^{-x/2}L_{n+\ell}^{[2\ell+1]}(x).

This equation arises in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom.

Physicists often use a definition for the Laguerre polynomials that is larger, by a factor of (n!), than the definition used here.

For further details, including the expressions for the first few polynomials, see Laguerre polynomials.

Hermite polynomials

The differential equation is

y'' - 2xy' + {\lambda}\,y = 0,\qquad \mathrm{with}\qquad\lambda = 2n.\,

This is Hermite's equation.

The second form of the differential equation is

(e^{-x^2}\,y')' + e^{-x^2}\,\lambda\,y = 0.\,

The third form is

(e^{-x^2/2}\,y)'' + ({\lambda}+1-x^2)(e^{-x^2/2}\,y) = 0.\,

The recurrence relation is

H_{n+1}(x) = 2x\,H_n(x) - 2n\,H_{n-1}(x).\,

Rodrigues' formula is

H_n(x) = (-1)^n\,e^{x^2} \  \frac{d^n}{dx^n}\left(e^{-x^2}\right).

The first few Hermite polynomials are

H_0(x) = 1\,
H_1(x) = 2x\,
H_2(x) = 4x^2-2\,
H_3(x) = 8x^3-12x\,
H_4(x) = 16x^4-48x^2+12\,

One can define the associated Hermite functions

{\psi}_n(x) = (h_n)^{-1/2}\,e^{-x^2/2}H_n(x).\,

Because the multiplier is proportional to the square root of the weight function, these functions are orthogonal over (-\infty, \infty) with no weight function.

The third form of the differential equation above, for the associated Hermite functions, is

\psi'' + ({\lambda}+1-x^2)\psi = 0.\,

The associated Hermite functions arise in many areas of mathematics and physics. In quantum mechanics, they are the solutions of Schrödinger's equation for the harmonic oscillator. They are also eigenfunctions (with eigenvalue (−i)n) of the continuous Fourier transform.

Many authors, particularly probabilists, use an alternate definition of the Hermite polynomials, with a weight function of e^{-x^2/2} instead of e^{-x^2}. If the notation He is used for these Hermite polynomials, and H for those above, then these may be characterized by

He_n(x) = 2^{-n/2}\,H_n\left(\frac{x}{\sqrt{2}}\right).

For further details, see Hermite polynomials.

Characterizations of classical orthogonal polynomials

There are several conditions that single out the classical orthogonal polynomials from the others.

The first condition was found by Sonine (and later by Hahn), who showed that (up to linear changes of variable) the classical orthogonal polynomials are the only ones such that their derivatives are also orthogonal polynomials.

Bochner characterized classical orthogonal polynomials in terms of their recurrence relations.

Tricomi characterized classical orthogonal polynmials as those that have a certain analogue of the Rodrigues formula.

Table of classical orthogonal polynomials

The following table summarises the properties of the classical orthogonal polynomials.[3]

Name, and conventional symbol Chebyshev, \ T_n Chebyshev
(second kind), \ U_n
Legendre, \ P_n Hermite, \ H_n
Limits of orthogonality[4] -1, 1\, -1, 1\, -1, 1\, -\infty, \infty
Weight, W(x)\, (1-x^2)^{-1/2}\, (1-x^2)^{1/2}\, 1\, e^{-x^2}
Standardization T_n(1)=1\, U_n(1)=n+1\, P_n(1)=1\, Lead term = 2^n\,
Square of norm [5] \left\{
\begin{matrix}
\pi   &:~n=0 \\
\pi/2 &:~n\ne 0
\end{matrix}\right.
\pi/2\, \frac{2}{2n+1} 2^n\,n!\,\sqrt{\pi}
Leading term [6] 2^{n-1}\, 2^n\, \frac{(2n)!}{2^n\,(n!)^2}\, 2^n\,
Second term, k'_n\, 0\, 0\, 0\, 0\,
Q\, 1-x^2\, 1-x^2\, 1-x^2\, 1\,
L\, -x\, -3x\, -2x\, -2x\,
R(x) =e^{\int \frac{L(x)}{Q(x)}\,dx} (1-x^2)^{1/2}\, (1-x^2)^{3/2}\, 1-x^2\, e^{-x^2}\,
Constant in diff. equation, {\lambda}_n\, n^2\, n(n+2)\, n(n+1)\, 2n\,
Constant in Rodrigues' formula, e_n\, (-2)^n\,\frac{\Gamma(n+1/2)}{\sqrt{\pi}}\, 2(-2)^n\,\frac{\Gamma(n+3/2)}{(n+1)\,\sqrt{\pi}}\, (-2)^n\,n!\, (-1)^n\,
Recurrence relation, a_n\, 2\, 2\, \frac{2n+1}{n+1}\, 2\,
Recurrence relation, b_n\, 0\, 0\, 0\, 0\,
Recurrence relation, c_n\, 1\, 1\, \frac{n}{n+1}\, 2n\,
Name, and conventional symbol Associated Laguerre, L_n^{(\alpha)} Laguerre, \ L_n
Limits of orthogonality 0, \infty\, 0, \infty\,
Weight, W(x)\, x^{\alpha}e^{-x}\, e^{-x}\,
Standardization Lead term = \frac{(-1)^n}{n!}\, Lead term = \frac{(-1)^n}{n!}\,
Square of norm, h_n\, \frac{\Gamma(n+\alpha+1)}{n!}\, 1\,
Leading term, k_n\, \frac{(-1)^n}{n!}\, \frac{(-1)^n}{n!}\,
Second term, k'_n\, \frac{(-1)^{n+1}(n+\alpha)}{(n-1)!}\, \frac{(-1)^{n+1}n}{(n-1)!}\,
Q\, x\, x\,
L\, \alpha+1-x\, 1-x\,
R(x) =e^{\int \frac{L(x)}{Q(x)}\,dx} x^{\alpha+1}\,e^{-x}\, x\,e^{-x}\,
Constant in diff. equation, {\lambda}_n\, n\, n\,
Constant in Rodrigues' formula, e_n\, n!\, n!\,
Recurrence relation, a_n\, \frac{-1}{n+1}\, \frac{-1}{n+1}\,
Recurrence relation, b_n\, \frac{2n+1+\alpha}{n+1}\, \frac{2n+1}{n+1}\,
Recurrence relation, c_n\, \frac{n+\alpha}{n+1}\, \frac{n}{n+1}\,
Name, and conventional symbol Gegenbauer, C_n^{(\alpha)} Jacobi, P_n^{(\alpha, \beta)}
Limits of orthogonality -1, 1\, -1, 1\,
Weight, W(x)\, (1-x^2)^{\alpha-1/2}\, (1-x)^\alpha(1+x)^\beta\,
Standardization C_n^{(\alpha)}(1)=\frac{\Gamma(n+2\alpha)}{n!\,\Gamma(2\alpha)}\, if \alpha\ne0 P_n^{(\alpha, \beta)}(1)=\frac{\Gamma(n+1+\alpha)}{n!\,\Gamma(1+\alpha)}\,
Square of norm, h_n\, \frac{\pi\,2^{1-2\alpha}\Gamma(n+2\alpha)}{n!(n+\alpha)(\Gamma(\alpha))^2} \frac{2^{\alpha+\beta+1}\,\Gamma(n\!+\!\alpha\!+\!1)\,\Gamma(n\!+\!\beta\!+\!1)}
{n!(2n\!+\!\alpha\!+\!\beta\!+\!1)\Gamma(n\!+\!\alpha\!+\!\beta\!+\!1)}
Leading term, k_n\, \frac{\Gamma(2n+2\alpha)\Gamma(1/2+\alpha)}{n!\,2^n\,\Gamma(2\alpha)\Gamma(n+1/2+\alpha)}\, \frac{\Gamma(2n+1+\alpha+\beta)}{n!\,2^n\,\Gamma(n+1+\alpha+\beta)}\,
Second term, k'_n\, 0\, \frac{(\alpha-\beta)\,\Gamma(2n+\alpha+\beta)}{(n-1)!\,2^n\,\Gamma(n+1+\alpha+\beta)}\,
Q\, 1-x^2\, 1-x^2\,
L\, -(2\alpha+1)\,x\, \beta-\alpha-(\alpha+\beta+2)\,x\,
R(x) =e^{\int \frac{L(x)}{Q(x)}\,dx} (1-x^2)^{\alpha+1/2}\, (1-x)^{\alpha+1}(1+x)^{\beta+1}\,
Constant in diff. equation, {\lambda}_n\, n(n+2\alpha)\, n(n+1+\alpha+\beta)\,
Constant in Rodrigues' formula, e_n\, \frac{(-2)^n\,n!\,\Gamma(2\alpha)\,\Gamma(n\!+\!1/2\!+\!\alpha)}
{\Gamma(n\!+\!2\alpha)\Gamma(\alpha\!+\!1/2)} (-2)^n\,n!\,
Recurrence relation, a_n\, \frac{2(n+\alpha)}{n+1}\, \frac{(2n+1+\alpha+\beta)(2n+2+\alpha+\beta)}{2(n+1)(n+1+\alpha+\beta)}
Recurrence relation, b_n\, 0\, \frac{({\alpha}^2-{\beta}^2)(2n+1+\alpha+\beta)}{2(n+1)(2n+\alpha+\beta)(n+1+\alpha+\beta)}
Recurrence relation, c_n\, \frac{n+2{\alpha}-1}{n+1}\, \frac{(n+\alpha)(n+\beta)(2n+2+\alpha+\beta)}{(n+1)(n+1+\alpha+\beta)(2n+\alpha+\beta)}

Notes

  1. ^ See Suetin (2001)
  2. ^ other conventions are also used; see Hermite polynomials.
  3. ^ See Abramowitz & Stegun (1965)
  4. ^ i.e. the edges of the support of the weight W.
  5. ^ h_n = \int P_n^2(x) W(x) dx
  6. ^ The leading coefficient kn of  P_n(x) = k_n x^n + k'_n x^{n-1} + \cdots + k^{(n)}

See also

References


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Orthogonal polynomials — In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the… …   Wikipedia

  • Orthogonal polynomials on the unit circle — In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane, for some probability measure on the unit circle. They were introduced …   Wikipedia

  • Kravchuk polynomials — or Krawtchouk polynomials are classical orthogonal polynomials associated with the binomial distribution, introduced by the Ukrainian mathematician Mikhail Kravchuk in 1929. [Sur une généralisation des polynomes d Hermite. Note de M.Krawtchouk, C …   Wikipedia

  • Hermite polynomials — In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical… …   Wikipedia

  • Koornwinder polynomials — In mathematics, Koornwinder polynomials are a family of orthogonal polynomials in several variables, named for their discoverer Tom H. Koornwinder, that generalize the Askey Wilson polynomials. They can also be viewed as Macdonald polynomials… …   Wikipedia

  • Legendre polynomials — Note: People sometimes refer to the more general associated Legendre polynomials as simply Legendre polynomials . In mathematics, Legendre functions are solutions to Legendre s differential equation::{d over dx} left [ (1 x^2) {d over dx} P n(x)… …   Wikipedia

  • Bernoulli polynomials — In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence… …   Wikipedia

  • List of real analysis topics — This is a list of articles that are considered real analysis topics. Contents 1 General topics 1.1 Limits 1.2 Sequences and Series 1.2.1 Summation Methods …   Wikipedia

  • Mehler–Heine formula — In mathematics, the Mehler–Heine formula introduced by Mehler (1868) and Heine (1861) describes the asymptotic behavior of the Legendre polynomials as the index tends to infinity, near the edges of the support of the weight. There are… …   Wikipedia

  • Hilbert space — For the Hilbert space filling curve, see Hilbert curve. Hilbert spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”