Classical orthogonal polynomials

In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and 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 to solve the moment problem by P. L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials.

For given polynomials ${\displaystyle Q,L:\mathbb {R} \to \mathbb {R} }$ and ${\displaystyle \forall \,n\in \mathbb {N} _{0}}$ the classical orthogonal polynomials ${\displaystyle f_{n}:\mathbb {R} \to \mathbb {R} }$ are characterized by being solutions of the differential equation

${\displaystyle Q(x)\,f_{n}^{\prime \prime }+L(x)\,f_{n}^{\prime }+\lambda _{n}f_{n}=0}$

with to be determined constants ${\displaystyle \lambda _{n}\in \mathbb {R} }$.

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.

In general, the orthogonal polynomials ${\displaystyle P_{n}}$ with respect to a weight ${\displaystyle W:\mathbb {R} \rightarrow \mathbb {R} ^{+}}$ satisfy

${\displaystyle {\begin{aligned}&\deg P_{n}=n~,\quad n=0,1,2,\ldots \\&\int P_{m}(x)\,P_{n}(x)\,W(x)\,dx=0~,\quad m\neq n~.\end{aligned}}}$

The relations above define ${\displaystyle P_{n}}$ up to multiplication by a number. Various normalisations are used to fix the constant, e.g.

${\displaystyle \int P_{n}(x)^{2}W(x)\,dx=1~.}$

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

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

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

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

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

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

${\displaystyle Q(x)\,f''+L(x)\,f'+\lambda f=0}$

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 as eigenvector/eigenvalue problems: Letting D be the differential operator, ${\displaystyle D(f)=Qf''+Lf'}$, 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 λ₀, λ₁, λ₂, ... that led to a series of polynomial solutions P₀, P₁, P₂, ... 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 P₀, P₁, P₂, ..., each P_n having degree n, and corresponding to a number λ_n.
• The interval of orthogonality is bounded by whatever roots Q has.
• The root of L is inside the interval of orthogonality.
• Letting ${\displaystyle R(x)=e^{\int {\frac {L(x)}{Q(x)}}\,dx}}$, the polynomials are orthogonal under the weight function ${\displaystyle W(x)={\frac {R(x)}{Q(x)}}}$
• 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).

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 ${\displaystyle m\neq n}$.

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 ${\displaystyle P_{n}^{(\alpha ,\beta )}}$. 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 ${\displaystyle [0,\infty )}$, and has Q = x. They can then be standardized into the Associated Laguerre polynomials ${\displaystyle L_{n}^{(\alpha )}}$. The plain Laguerre polynomials ${\displaystyle \ L_{n}}$ 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 ${\displaystyle (-\infty ,\infty )}$, and has Q = 1 and L(0) = 0. They can then be standardized into the Hermite polynomials ${\displaystyle H_{n}}$.

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.

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.

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 polynomials as those that have a certain analogue of the Rodrigues formula.

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

Name, and conventional symbol	Chebyshev, ${\displaystyle \ T_{n}}$	Chebyshev (second kind), ${\displaystyle \ U_{n}}$	Legendre, ${\displaystyle \ P_{n}}$	Hermite, ${\displaystyle \ H_{n}}$
Limits of orthogonality[4]	${\displaystyle -1,1}$	${\displaystyle -1,1}$	${\displaystyle -1,1}$	${\displaystyle -\infty ,\infty }$
Weight, ${\displaystyle W(x)}$	${\displaystyle (1-x^{2})^{-1/2}}$	${\displaystyle (1-x^{2})^{1/2}}$	${\displaystyle 1}$	${\displaystyle e^{-x^{2}}}$
Standardization	${\displaystyle T_{n}(1)=1}$	${\displaystyle U_{n}(1)=n+1}$	${\displaystyle P_{n}(1)=1}$	Lead term ${\displaystyle =2^{n}}$
Square of norm [5]	${\displaystyle \left\{{\begin{matrix}\pi &:~n=0\\\pi /2&:~n\neq 0\end{matrix}}\right.}$	${\displaystyle \pi /2}$	${\displaystyle {\frac {2}{2n+1}}}$	${\displaystyle 2^{n}\,n!\,{\sqrt {\pi }}}$
Leading term [6]	${\displaystyle 2^{n-1}}$	${\displaystyle 2^{n}}$	${\displaystyle {\frac {(2n)!}{2^{n}\,(n!)^{2}}}}$	${\displaystyle 2^{n}}$
Second term, ${\displaystyle k'_{n}}$	${\displaystyle 0}$	${\displaystyle 0}$	${\displaystyle 0}$	${\displaystyle 0}$
${\displaystyle Q}$	${\displaystyle 1-x^{2}}$	${\displaystyle 1-x^{2}}$	${\displaystyle 1-x^{2}}$	${\displaystyle 1}$
${\displaystyle L}$	${\displaystyle -x}$	${\displaystyle -3x}$	${\displaystyle -2x}$	${\displaystyle -2x}$
${\displaystyle R(x)=e^{\int {\frac {L(x)}{Q(x)}}\,dx}}$	${\displaystyle (1-x^{2})^{1/2}}$	${\displaystyle (1-x^{2})^{3/2}}$	${\displaystyle 1-x^{2}}$	${\displaystyle e^{-x^{2}}}$
Constant in diff. equation, ${\displaystyle \lambda _{n}}$	${\displaystyle n^{2}}$	${\displaystyle n(n+2)}$	${\displaystyle n(n+1)}$	${\displaystyle 2n}$
Constant in Rodrigues' formula, ${\displaystyle e_{n}}$	${\displaystyle (-2)^{n}\,{\frac {\Gamma (n+1/2)}{\sqrt {\pi }}}}$	${\displaystyle 2(-2)^{n}\,{\frac {\Gamma (n+3/2)}{(n+1)\,{\sqrt {\pi }}}}}$	${\displaystyle (-2)^{n}\,n!}$	${\displaystyle (-1)^{n}}$
Recurrence relation, ${\displaystyle a_{n}}$	${\displaystyle 2}$	${\displaystyle 2}$	${\displaystyle {\frac {2n+1}{n+1}}}$	${\displaystyle 2}$
Recurrence relation, ${\displaystyle b_{n}}$	${\displaystyle 0}$	${\displaystyle 0}$	${\displaystyle 0}$	${\displaystyle 0}$
Recurrence relation, ${\displaystyle c_{n}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle {\frac {n}{n+1}}}$	${\displaystyle 2n}$

Name, and conventional symbol	Associated Laguerre, ${\displaystyle L_{n}^{(\alpha )}}$	Laguerre, ${\displaystyle \ L_{n}}$
Limits of orthogonality	${\displaystyle 0,\infty }$	${\displaystyle 0,\infty }$
Weight, ${\displaystyle W(x)}$	${\displaystyle x^{\alpha }e^{-x}}$	${\displaystyle e^{-x}}$
Standardization	Lead term ${\displaystyle ={\frac {(-1)^{n}}{n!}}}$	Lead term ${\displaystyle ={\frac {(-1)^{n}}{n!}}}$
Square of norm, ${\displaystyle h_{n}}$	${\displaystyle {\frac {\Gamma (n+\alpha +1)}{n!}}}$	${\displaystyle 1}$
Leading term, ${\displaystyle k_{n}}$	${\displaystyle {\frac {(-1)^{n}}{n!}}}$	${\displaystyle {\frac {(-1)^{n}}{n!}}}$
Second term, ${\displaystyle k'_{n}}$	${\displaystyle {\frac {(-1)^{n+1}(n+\alpha )}{(n-1)!}}}$	${\displaystyle {\frac {(-1)^{n+1}n}{(n-1)!}}}$
${\displaystyle Q}$	${\displaystyle x}$	${\displaystyle x}$
${\displaystyle L}$	${\displaystyle \alpha +1-x}$	${\displaystyle 1-x}$
${\displaystyle R(x)=e^{\int {\frac {L(x)}{Q(x)}}\,dx}}$	${\displaystyle x^{\alpha +1}\,e^{-x}}$	${\displaystyle x\,e^{-x}}$
Constant in diff. equation, ${\displaystyle \lambda _{n}}$	${\displaystyle n}$	${\displaystyle n}$
Constant in Rodrigues' formula, ${\displaystyle e_{n}}$	${\displaystyle n!}$	${\displaystyle n!}$
Recurrence relation, ${\displaystyle a_{n}}$	${\displaystyle {\frac {-1}{n+1}}}$	${\displaystyle {\frac {-1}{n+1}}}$
Recurrence relation, ${\displaystyle b_{n}}$	${\displaystyle {\frac {2n+1+\alpha }{n+1}}}$	${\displaystyle {\frac {2n+1}{n+1}}}$
Recurrence relation, ${\displaystyle c_{n}}$	${\displaystyle {\frac {n+\alpha }{n+1}}}$	${\displaystyle {\frac {n}{n+1}}}$

Name, and conventional symbol	Gegenbauer, ${\displaystyle C_{n}^{(\alpha )}}$	Jacobi, ${\displaystyle P_{n}^{(\alpha ,\beta )}}$
Limits of orthogonality	${\displaystyle -1,1}$	${\displaystyle -1,1}$
Weight, ${\displaystyle W(x)}$	${\displaystyle (1-x^{2})^{\alpha -1/2}}$	${\displaystyle (1-x)^{\alpha }(1+x)^{\beta }}$
Standardization	${\displaystyle C_{n}^{(\alpha )}(1)={\frac {\Gamma (n+2\alpha )}{n!\,\Gamma (2\alpha )}}}$ if ${\displaystyle \alpha \neq 0}$	${\displaystyle P_{n}^{(\alpha ,\beta )}(1)={\frac {\Gamma (n+1+\alpha )}{n!\,\Gamma (1+\alpha )}}}$
Square of norm, ${\displaystyle h_{n}}$	${\displaystyle {\frac {\pi \,2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )(\Gamma (\alpha ))^{2}}}}$	${\displaystyle {\frac {2^{\alpha +\beta +1}\,\Gamma (n\!+\!\alpha \!+\!1)\,\Gamma (n\!+\!\beta \!+\!1)}{n!(2n\!+\!\alpha \!+\!\beta \!+\!1)\Gamma (n\!+\!\alpha \!+\!\beta \!+\!1)}}}$
Leading term, ${\displaystyle k_{n}}$	${\displaystyle {\frac {\Gamma (2n+2\alpha )\Gamma (1/2+\alpha )}{n!\,2^{n}\,\Gamma (2\alpha )\Gamma (n+1/2+\alpha )}}}$	${\displaystyle {\frac {\Gamma (2n+1+\alpha +\beta )}{n!\,2^{n}\,\Gamma (n+1+\alpha +\beta )}}}$
Second term, ${\displaystyle k'_{n}}$	${\displaystyle 0}$	${\displaystyle {\frac {(\alpha -\beta )\,\Gamma (2n+\alpha +\beta )}{(n-1)!\,2^{n}\,\Gamma (n+1+\alpha +\beta )}}}$
${\displaystyle Q}$	${\displaystyle 1-x^{2}}$	${\displaystyle 1-x^{2}}$
${\displaystyle L}$	${\displaystyle -(2\alpha +1)\,x}$	${\displaystyle \beta -\alpha -(\alpha +\beta +2)\,x}$
${\displaystyle R(x)=e^{\int {\frac {L(x)}{Q(x)}}\,dx}}$	${\displaystyle (1-x^{2})^{\alpha +1/2}}$	${\displaystyle (1-x)^{\alpha +1}(1+x)^{\beta +1}}$
Constant in diff. equation, ${\displaystyle \lambda _{n}}$	${\displaystyle n(n+2\alpha )}$	${\displaystyle n(n+1+\alpha +\beta )}$
Constant in Rodrigues' formula, ${\displaystyle e_{n}}$	${\displaystyle {\frac {(-2)^{n}\,n!\,\Gamma (2\alpha )\,\Gamma (n\!+\!1/2\!+\!\alpha )}{\Gamma (n\!+\!2\alpha )\Gamma (\alpha \!+\!1/2)}}}$	${\displaystyle (-2)^{n}\,n!}$
Recurrence relation, ${\displaystyle a_{n}}$	${\displaystyle {\frac {2(n+\alpha )}{n+1}}}$	${\displaystyle {\frac {(2n+1+\alpha +\beta )(2n+2+\alpha +\beta )}{2(n+1)(n+1+\alpha +\beta )}}}$
Recurrence relation, ${\displaystyle b_{n}}$	${\displaystyle 0}$	${\displaystyle {\frac {({\alpha }^{2}-{\beta }^{2})(2n+1+\alpha +\beta )}{2(n+1)(2n+\alpha +\beta )(n+1+\alpha +\beta )}}}$
Recurrence relation, ${\displaystyle c_{n}}$	${\displaystyle {\frac {n+2{\alpha }-1}{n+1}}}$	${\displaystyle {\frac {(n+\alpha )(n+\beta )(2n+2+\alpha +\beta )}{(n+1)(n+1+\alpha +\beta )(2n+\alpha +\beta )}}}$

• Appell sequence
• Askey scheme of hypergeometric orthogonal polynomials
• Polynomial sequences of binomial type
• Biorthogonal polynomials
• Generalized Fourier series
• Secondary measure
• Sheffer sequence
• Umbral calculus

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 773. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
• Andrews, George E.; Askey, Richard (1985). "Classical orthogonal polynomials". In Brezinski, C.; Draux, A.; Magnus, Alphonse P.; Maroni, Pascal; Ronveaux, A. (eds.). Polynômes orthogonaux et applications. Proceedings of the Laguerre symposium held at Bar-le-Duc, October 15–18, 1984. Lecture Notes in Math. Vol. 1171. Berlin, New York: Springer-Verlag. pp. 36–62. doi:10.1007/BFb0076530. ISBN 978-3-540-16059-5. MR 0838970.
• Chihara, Theodore Seio (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. ISBN 0-677-04150-0.
• Foncannon, J. J.; Foncannon, J. J.; Pekonen, Osmo (2008). "Review of Classical and quantum orthogonal polynomials in one variable by Mourad Ismail". The Mathematical Intelligencer. 30. Springer New York: 54–60. doi:10.1007/BF02985757. ISSN 0343-6993. S2CID 118133026.
• Ismail, Mourad E. H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge: Cambridge Univ. Press. ISBN 0-521-78201-5.
• Jackson, Dunham (2004) [1941]. Fourier Series and Orthogonal Polynomials. New York: Dover. ISBN 0-486-43808-2.
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Suetin, P. K. (2001) [1994], "Classical orthogonal polynomials", Encyclopedia of Mathematics, EMS Press
• Szegő, Gábor (1939). Orthogonal Polynomials. Colloquium Publications. Vol. XXIII. American Mathematical Society. ISBN 978-0-8218-1023-1. MR 0372517.