Editing Classical orthogonal polynomials (section)

== 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 [&minus;1,&nbsp;1], and has ''Q'' = 1&nbsp;&minus;&nbsp;''x''<sup>2</sup>.  They can then be standardized into the '''Jacobi polynomials''' <math>P_n^{(\alpha, \beta)}</math>.  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 <math>[0, \infty)</math>, and has ''Q'' = ''x''.  They can then be standardized into the '''Associated Laguerre polynomials''' <math>L_n^{(\alpha)}</math>.  The plain '''Laguerre polynomials''' <math>\ L_n</math> 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 <math>(-\infty, \infty)</math>, and has Q = 1 and L(0) = 0.  They can then be standardized into the '''Hermite polynomials''' <math>H_n</math>.

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 [&minus;1,&nbsp;1], still have two parameters to be determined.
They are <math>\alpha</math> and <math>\beta</math> in the Jacobi polynomials,
written <math>P_n^{(\alpha, \beta)}</math>.  We have <math>Q(x) = 1-x^2</math> and
<math>L(x) = \beta-\alpha-(\alpha+\beta+2)\, x</math>.
Both <math>\alpha</math> and <math>\beta</math> are required to be greater than &minus;1.
(This puts the root of L inside the interval of orthogonality.)

When <math>\alpha</math> and <math>\beta</math> are not equal, these polynomials
are not symmetrical about ''x'' = 0.

The differential equation

:<math>(1-x^2)\,y'' + (\beta-\alpha-[\alpha+\beta+2]\,x)\,y' + \lambda \,y = 0\qquad \text{with}\qquad\lambda = n(n+1+\alpha+\beta)</math>

is '''Jacobi's equation'''.

For further details, see [[Jacobi polynomials]].

=== Gegenbauer polynomials ===
When one sets the parameters <math>\alpha</math> and <math>\beta</math> in the Jacobi polynomials equal to each other, one obtains the '''Gegenbauer''' or '''ultraspherical''' polynomials.  They are written <math>C_n^{(\alpha)}</math>, and defined as

:<math>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)}(x).</math>

We have <math>Q(x) = 1-x^2</math> and
<math>L(x) = -(2\alpha+1)\, x</math>.
The parameter <math>\alpha</math> is required to be greater than&nbsp;&minus;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 <math>C_n^{(0)}(1) = \frac{2}{n}</math> instead of the value given in the table.)

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

:<math>C_n^{(\alpha+1)}(x) = \frac{1}{2\alpha}\! \  \frac{d}{dx}C_{n+1}^{(\alpha)}(x)</math>

or, more generally:

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

All the other classical Jacobi-like polynomials (Legendre, etc.) are special cases of the Gegenbauer polynomials, obtained by choosing a value of <math>\alpha</math> and choosing a standardization.

For further details, see [[Gegenbauer polynomials]].

=== Legendre polynomials ===
The differential equation is

:<math>(1-x^2)\,y'' - 2x\,y' + \lambda \,y = 0\qquad \text{with}\qquad\lambda = n(n+1).</math>

This is '''Legendre's equation'''.

The second form of the differential equation is:

:<math>\frac{d}{dx}[(1-x^2)\,y'] + \lambda\,y = 0.</math>

The [[recurrence relation]] is

:<math>(n+1)\,P_{n+1}(x) = (2n+1)x\,P_n(x) - n\,P_{n-1}(x).</math>

A mixed recurrence is

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

Rodrigues' formula is

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

For further details, see [[Legendre polynomials]].

==== Associated Legendre polynomials ====
The [[Associated Legendre polynomials]], denoted
<math>P_\ell^{(m)}(x)</math> where <math>\ell</math> and <math>m</math> are integers with <math>0 \leqslant m  \leqslant \ell</math>, are defined as

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

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

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

They have a recurrence relation:

:<math>(\ell+1-m)\,P_{\ell+1}^{(m)}(x) = (2\ell+1)x\,P_\ell^{(m)}(x) - (\ell+m)\,P_{\ell-1}^{(m)}(x).</math>

For fixed ''m'', the sequence <math>P_m^{(m)}, P_{m+1}^{(m)}, P_{m+2}^{(m)}, \dots</math> are orthogonal over [&minus;1,&nbsp;1], with weight&nbsp;1.

For given ''m'', <math>P_\ell^{(m)}(x)</math> are the solutions of

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

=== Chebyshev polynomials ===
The differential equation is

:<math>(1-x^2)\,y'' - x\,y' + \lambda \,y = 0\qquad \text{with}\qquad\lambda = n^2.</math>

This is '''[[Chebyshev equation|Chebyshev's equation]]'''.

The recurrence relation is

:<math>T_{n+1}(x) = 2x\,T_n(x) - T_{n-1}(x).</math>

Rodrigues' formula is

:<math>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).</math>

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

:<math>T_n(x) = \cos(n\,\arccos(x)).</math>

(To prove it, use the recurrence formula.)

This means that all their local minima and maxima have values of &minus;1 and +1, that is, the polynomials are "level".  Because of this, expansion of functions in terms of Chebyshev polynomials is sometimes used for [[approximation theory|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,&nbsp;1] or [&minus;2,&nbsp;2].

There are also '''Chebyshev polynomials of the second kind''', denoted <math>U_n</math>

We have:

:<math>U_n = \frac{1}{n+1}\,T_{n+1}'.</math>

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 <math>L_n^{(\alpha)}</math>.  There is a parameter <math>\alpha</math>, which can be any real number strictly greater than &minus;1.  The parameter is put in parentheses to avoid confusion with an exponent.  The plain Laguerre polynomials are simply the <math>\alpha = 0</math> version of these:

:<math>L_n(x) = L_n^{(0)}(x).</math>

The differential equation is

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

This is '''Laguerre's equation'''.

The second form of the differential equation is

:<math>(x^{\alpha+1}\,e^{-x}\, y')' + \lambda \,x^\alpha \,e^{-x}\,y = 0.</math>

The recurrence relation is

:<math>(n+1)\,L_{n+1}^{(\alpha)}(x) = (2n+1+\alpha-x)\,L_n^{(\alpha)}(x) - (n+\alpha)\,L_{n-1}^{(\alpha)}(x).</math>

Rodrigues' formula is

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

The parameter <math>\alpha</math> is closely related to the derivatives of <math>L_n^{(\alpha)}</math>:

:<math>L_n^{(\alpha+1)}(x) = - \frac{d}{dx}L_{n+1}^{(\alpha)}(x)</math>

or, more generally:

:<math>L_n^{(\alpha+m)}(x) = (-1)^m L_{n+m}^{(\alpha)[m]}(x).</math>

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

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

is a solution of

:<math>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}. </math>

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

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

is a solution of

:<math>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.</math>

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

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

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 <math>(n!)</math>, 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

:<math>y'' - 2xy' + \lambda \,y = 0,\qquad \text{with}\qquad\lambda = 2n.</math>

This is '''Hermite's equation'''.

The second form of the differential equation is

:<math>(e^{-x^2}\,y')' + e^{-x^2}\,\lambda\,y = 0.</math>

The third form is

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

The recurrence relation is

:<math>H_{n+1}(x) = 2x\,H_n(x) - 2n\,H_{n-1}(x).</math>

Rodrigues' formula is

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

The first few Hermite polynomials are

:<math>H_0(x) = 1</math>

:<math>H_1(x) = 2x</math>

:<math>H_2(x) = 4x^2-2</math>

:<math>H_3(x) = 8x^3-12x</math>

:<math>H_4(x) = 16x^4-48x^2+12</math>

One can define the '''associated Hermite functions'''

: <math> \psi_n(x) = (h_n)^{-1/2}\,e^{-x^2/2}H_n(x).</math>

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

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

:<math>\psi'' + (\lambda +1-x^2)\psi = 0.</math>

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 (&minus;''i'' <sup>''n''</sup>) of the [[continuous Fourier transform]].

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

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

For further details, see [[Hermite polynomials]].