Classical orthogonal polynomials

For ${\displaystyle \alpha ,\,\beta >-1}$ the Jacobi polynomials are given by the formula

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

They are normalised (standardized) by

${\displaystyle P_{n}^{(\alpha ,\beta )}(1)={n+\alpha \choose n},}$

and satisfy the orthogonality condition

${\displaystyle {\begin{aligned}&\int _{-1}^{1}(1-x)^{\alpha }(1+x)^{\beta }P_{m}^{(\alpha ,\beta )}(x)P_{n}^{(\alpha ,\beta )}(x)\;dx\\={}&{\frac {2^{\alpha +\beta +1}}{2n+\alpha +\beta +1}}{\frac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{\Gamma (n+\alpha +\beta +1)n!}}\delta _{nm}.\end{aligned}}}$

The Jacobi polynomials are solutions to the differential equation

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

The Jacobi polynomials with ${\displaystyle \alpha =\beta }$ are called the Gegenbauer polynomials (with parameter ${\displaystyle \gamma =\alpha +1/2}$)

For ${\displaystyle \alpha =\beta =0}$, these are called the Legendre polynomials (for which the interval of orthogonality is [−1, 1] and the weight function is simply 1):

${\displaystyle P_{0}(x)=1,\,P_{1}(x)=x,\,P_{2}(x)={\frac {3x^{2}-1}{2}},\,P_{3}(x)={\frac {5x^{3}-3x}{2}},\ldots }$

For ${\displaystyle \alpha =\beta =\pm 1/2}$, one obtains the Chebyshev polynomials (of the second and first kind, respectively).

The Hermite polynomials are defined by[2]

${\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}=e^{x^{2}/2}{\bigg (}x-{\frac {d}{dx}}{\bigg )}^{n}e^{-x^{2}/2}}$

They satisfy the orthogonality condition

${\displaystyle \int _{-\infty }^{\infty }H_{n}(x)H_{m}(x)e^{-x^{2}}\,dx={\sqrt {\pi }}2^{n}n!\delta _{mn}~,}$

and the differential equation

${\displaystyle y''-2xy'+2n\,y=0~.}$

The generalised Laguerre polynomials are defined by

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

(the classical Laguerre polynomials correspond to ${\displaystyle \alpha =0}$.)

They satisfy the orthogonality relation

${\displaystyle \int _{0}^{\infty }x^{\alpha }e^{-x}L_{n}^{(\alpha )}(x)L_{m}^{(\alpha )}(x)\,dx={\frac {\Gamma (n+\alpha +1)}{n!}}\delta _{n,m}~,}$

and the differential equation

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

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

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

${\displaystyle P_{n}(x)={\frac {1}{{e_{n}}W(x)}}\ {\frac {d^{n}}{dx^{n}}}\left(W(x)[Q(x)]^{n}\right)}$

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

Under the assumptions of the preceding section, we have

${\displaystyle \lambda _{n}=-n\left({\frac {n-1}{2}}Q''+L'\right).}$

(Since Q is quadratic and L is linear, ${\displaystyle Q''}$ and ${\displaystyle L'}$ are constants, so these are just numbers.)

Let

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

Then

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

Now multiply the differential equation

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

by R/Q, getting

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

or

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

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

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

Then

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

Now multiply the differential equation

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

by S/Q, getting

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

or

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

But ${\displaystyle (S\,y)''=S\,y''+2\,S'\,y'+S''\,y}$, so

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

or, letting u = Sy,

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

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

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

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.

• ${\displaystyle P_{n}^{[r]}=aP_{n+1}^{[r+1]}+bP_{n}^{[r+1]}+cP_{n-1}^{[r+1]}}$
• ${\displaystyle P_{n}^{[r]}=(ax+b)P_{n}^{[r+1]}+cP_{n-1}^{[r+1]}}$
• ${\displaystyle QP_{n}^{[r+1]}=(ax+b)P_{n}^{[r]}+cP_{n-1}^{[r]}}$

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:

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

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

${\displaystyle Q{\ddot {f}}_{n}+L{\dot {f}}_{n}+\lambda _{n}f_{n}=0}$

multiplying by ${\displaystyle (R/Q)f_{m}}$ yields

${\displaystyle Rf_{m}{\ddot {f}}_{n}+{\frac {R}{Q}}Lf_{m}{\dot {f}}_{n}+{\frac {R}{Q}}\lambda _{n}f_{m}f_{n}=0}$

and reversing the subscripts yields

${\displaystyle Rf_{n}{\ddot {f}}_{m}+{\frac {R}{Q}}Lf_{n}{\dot {f}}_{m}+{\frac {R}{Q}}\lambda _{m}f_{n}f_{m}=0}$

subtracting and integrating:

${\displaystyle \int _{a}^{b}\left[R(f_{m}{\ddot {f}}_{n}-f_{n}{\ddot {f}}_{m})+{\frac {R}{Q}}L(f_{m}{\dot {f}}_{n}-f_{n}{\dot {f}}_{m})\right]\,dx+(\lambda _{n}-\lambda _{m})\int _{a}^{b}{\frac {R}{Q}}f_{m}f_{n}\,dx=0}$

but it can be seen that

${\displaystyle {\frac {d}{dx}}\left[R(f_{m}{\dot {f}}_{n}-f_{n}{\dot {f}}_{m})\right]=R(f_{m}{\ddot {f}}_{n}-f_{n}{\ddot {f}}_{m})\,\,+\,\,R{\frac {L}{Q}}(f_{m}{\dot {f}}_{n}-f_{n}{\dot {f}}_{m})}$

so that:

${\displaystyle \left[R(f_{m}{\dot {f}}_{n}-f_{n}{\dot {f}}_{m})\right]_{a}^{b}\,\,+\,\,(\lambda _{n}-\lambda _{m})\int _{a}^{b}{\frac {R}{Q}}f_{m}f_{n}\,dx=0}$

If the polynomials f are such that the term on the left is zero, and ${\displaystyle \lambda _{m}\neq \lambda _{n}}$ for ${\displaystyle m\neq n}$, then the orthogonality relationship will hold:

${\displaystyle \int _{a}^{b}{\frac {R}{Q}}f_{m}f_{n}\,dx=0}$

for ${\displaystyle m\neq n}$.

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 ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ in the Jacobi polynomials, written ${\displaystyle P_{n}^{(\alpha ,\beta )}}$. We have ${\displaystyle Q(x)=1-x^{2}}$ and ${\displaystyle L(x)=\beta -\alpha -(\alpha +\beta +2)\,x}$. Both ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are required to be greater than −1. (This puts the root of L inside the interval of orthogonality.)

When ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are not equal, these polynomials are not symmetrical about x = 0.

The differential equation

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

is Jacobi's equation.

For further details, see Jacobi polynomials.

When one sets the parameters ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ in the Jacobi polynomials equal to each other, one obtains the Gegenbauer or ultraspherical polynomials. They are written ${\displaystyle C_{n}^{(\alpha )}}$, and defined as

${\displaystyle C_{n}^{(\alpha )}(x)={\frac {\Gamma (2\alpha \!+\!n)\,\Gamma (\alpha \!+\!1/2)}{\Gamma (2\alpha )\,\Gamma (\alpha \!+\!n\!+\!1/2)}}\!\ P_{n}^{(\alpha -1/2,\alpha -1/2)}(x).}$

We have ${\displaystyle Q(x)=1-x^{2}}$ and ${\displaystyle L(x)=-(2\alpha +1)\,x}$. The parameter ${\displaystyle \alpha }$ 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 ${\displaystyle C_{n}^{(0)}(1)={\frac {2}{n}}}$ instead of the value given in the table.)

Ignoring the above considerations, the parameter ${\displaystyle \alpha }$ is closely related to the derivatives of ${\displaystyle C_{n}^{(\alpha )}}$:

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

or, more generally:

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

All the other classical Jacobi-like polynomials (Legendre, etc.) are special cases of the Gegenbauer polynomials, obtained by choosing a value of ${\displaystyle \alpha }$ and choosing a standardization.

For further details, see Gegenbauer polynomials.

The differential equation is

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

This is Legendre's equation.

The second form of the differential equation is:

${\displaystyle {\frac {d}{dx}}[(1-x^{2})\,y']+\lambda \,y=0.}$

The recurrence relation is

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

A mixed recurrence is

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

Rodrigues' formula is

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

For further details, see Legendre polynomials.

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

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

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

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

They have a recurrence relation:

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

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

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

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

The differential equation is

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

This is Chebyshev's equation.

The recurrence relation is

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

Rodrigues' formula is

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

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

${\displaystyle T_{n}(x)=\cos(n\,\arccos(x)).}$

(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 ${\displaystyle U_{n}}$

We have:

${\displaystyle U_{n}={\frac {1}{n+1}}\,T_{n+1}'.}$

For further details, including the expressions for the first few polynomials, see Chebyshev 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 ${\displaystyle L_{n}^{(\alpha )}}$. There is a parameter ${\displaystyle \alpha }$, 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 ${\displaystyle \alpha =0}$ version of these:

${\displaystyle L_{n}(x)=L_{n}^{(0)}(x).}$

The differential equation is

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

This is Laguerre's equation.

The second form of the differential equation is

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

The recurrence relation is

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

Rodrigues' formula is

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

The parameter ${\displaystyle \alpha }$ is closely related to the derivatives of ${\displaystyle L_{n}^{(\alpha )}}$:

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

or, more generally:

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

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

${\displaystyle u=x^{\frac {\alpha -1}{2}}e^{-x/2}L_{n}^{(\alpha )}(x)}$

is a solution of

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

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

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

is a solution of

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

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

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

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 ${\displaystyle (n!)}$, than the definition used here.

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

The differential equation is

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

This is Hermite's equation.

The second form of the differential equation is

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

The third form is

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

The recurrence relation is

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

Rodrigues' formula is

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

The first few Hermite polynomials are

${\displaystyle H_{0}(x)=1}$

${\displaystyle H_{1}(x)=2x}$

${\displaystyle H_{2}(x)=4x^{2}-2}$

${\displaystyle H_{3}(x)=8x^{3}-12x}$

${\displaystyle H_{4}(x)=16x^{4}-48x^{2}+12}$

One can define the associated Hermite functions

${\displaystyle \psi _{n}(x)=(h_{n})^{-1/2}\,e^{-x^{2}/2}H_{n}(x).}$

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

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

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

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 ${\displaystyle e^{-x^{2}/2}}$ instead of ${\displaystyle e^{-x^{2}}}$. If the notation He is used for these Hermite polynomials, and H for those above, then these may be characterized by

${\displaystyle He_{n}(x)=2^{-n/2}\,H_{n}\left({\frac {x}{\sqrt {2}}}\right).}$

For further details, see Hermite polynomials.