Classical orthogonal polynomials

In mathematics, an orthogonal polynomial sequence is an infinite sequence of real polynomials

$p_{0},\ p_{1},\ p_{2},\ \ldots$

of one variable x, in which each p_n has degree n, and such that any two different polynomials in the sequence are orthogonal to each other under a particular version of the L² inner product.

The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P. L. Chebyshev and was kept on by A.A. Markov and T.J. Stieltjes and by a few other mathematicians. Since then, applications have been developed in many areas of mathematics and physics.

Definition

The definition of orthogonal polynomials hinges on an inner product, defined as follows. Let $[x_{1},x_{2}]$ be an interval in the real line (where $x_{1}=-\infty$ and $x_{2}=\infty$ are allowed). This is called the interval of orthogonality. Let

$W:[x_{1},x_{2}]\to \mathbb {R}$

be a function on the interval, that is strictly positive on the interior $(x_{1},x_{2})$, but which may be zero or go to infinity at the end points. Additionally, W must satisfy the requirement that, for any polynomial $f$, the integral

$\int _{x_{1}}^{x_{2}}f(x)W(x)\;dx$

is finite. Such a W is called a weight function.

Given any $x_{1}$, $x_{2}$, and W as above, define an operation on pairs of polynomials f and g by

$\langle f,g\rangle =\int _{x_{1}}^{x_{2}}f(x)g(x)W(x)\;dx.$

This operation is an inner product on the vector space of all polynomials. It induces a notion of orthogonality in the usual way, namely that two polynomials are orthogonal if their inner product is zero.

A sequence of orthogonal polynomials, then, is a sequence of polynomials

$p_{0},\ p_{1},\ p_{2},\ \ldots$

such that $p_{n}$ has degree n and all members of the sequence are orthogonal to each other — for all $m\neq n$,

$\langle p_{m},p_{n}\rangle =0.$

In other words, a sequence of orthogonal polynomials is an orthogonal basis for the (infinite-dimensional) vector space of all polynomials, with the extra requirement that $p_{n}$ has degree n.

Standardization

The chosen inner product induces a norm on polynomials in the usual way:

$||f||=\langle f,f\rangle ^{1/2}.$

When making an orthogonal basis, one may be tempted to make an orthonormal basis, that is, one in which all basis elements have norm 1. For polynomials, this would often result in ugly square roots in the coefficients. Instead, polynomials are often scaled in a way that mathematicians agree on, that makes the coefficients and other formulas simpler. This is called standardization. The "classical" polynomials listed below have been standardized, typically by setting their leading coefficients to some specific quantity, or by setting a specific value for the polynomial. This standardization has no mathematical significance; it is just a convention. Standardization also involves scaling the weight function in an agreed-upon way.

Denote by $h_{n}$ the square of the norm of $p_{n}$:

$h_{n}=\langle p_{n},p_{n}\rangle .$

The values of $h_{n}$ for the standardized classical polynomials are listed in the table below. In this notation,

$\langle p_{m},p_{n}\rangle =\delta _{mn}{\sqrt {h_{m}h_{n}}},$

where δ_mn is the Kronecker delta.

Example: Legendre polynomials

The simplest orthogonal polynomials are the Legendre polynomials, for which the interval of orthogonality is [−1, 1] and the weight function is simply 1:

$P_{0}(x)=1,\,$

$P_{1}(x)=x,\,$

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

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

$P_{4}(x)={\frac {35x^{4}-30x^{2}+3}{8}},\,$

$\vdots$

These are all orthogonal over [−1, 1]; whenever $m\neq n$,

$\int _{-1}^{1}P_{m}(x)P_{n}(x)\,dx=0.$

The Legendre polynomials are standardized so that $P_{n}(1)=1$ for all n.

General properties of orthogonal polynomial sequences

All orthogonal polynomial sequences have a number of elegant and fascinating properties. Before proceeding with them:

Lemma 1: Given an orthogonal polynomial sequence $\ p_{i}(x)$, any n^th-degree polynomial S(x) can be expanded in terms of $p_{0},\dots ,p_{n}$. That is, there are coefficients ${\alpha }_{0},\dots ,{\alpha }_{n}$ such that

$S(x)=\sum _{i=0}^{n}{\alpha }_{i}\ p_{i}(x).$

Proof by mathematical induction. Choose $\ {\alpha }_{n}$ so that the $\ x^{n}$ term of S(x) matches that of $\ {\alpha }_{n}P_{n}(x)$. Then $\ S(x)-{\alpha }_{n}P_{n}(x)$ is an (n − 1)th-degree polynomial. Continue downward.

Lemma 2: Given an orthogonal polynomial sequence, each of its polynomials is orthogonal to any polynomial of strictly lower degree.

Proof: Given n, any polynomial of degree n − 1 or lower can be expanded in terms of $p_{0},\dots ,p_{n-1}$. $p_{n}\,$ is orthogonal to each of them.

Recurrence relations

Any orthogonal sequence has a recurrence formula relating any three consecutive polynomials in the sequence:

$p_{n+1}\ =\ (a_{n}x+b_{n})\ p_{n}\ -\ c_{n}\ p_{n-1}.$

The coefficients a, b, and c depend on n, as well as the standardization. (proof)

The values of $a_{n}$, $b_{n}$ and $c_{n}$ can be worked out directly. Let $k_{j}$ and $k_{j}'$ be the first and second coefficients of $p_{j}$:

$p_{j}(x)=k_{j}x^{j}+k_{j}'x^{j-1}+\cdots$

and $h_{j}$ be the inner product of $p_{j}$ with itself:

$h_{j}\ =\ \langle p_{j},\ p_{j}\rangle .$

We have

$a_{n}={\frac {k_{n+1}}{k_{n}}},\qquad b_{n}=a_{n}\left({\frac {k_{n+1}'}{k_{n+1}}}-{\frac {k_{n}'}{k_{n}}}\right),\qquad c_{n}=a_{n}\left({\frac {k_{n-1}h_{n}}{k_{n}h_{n-1}}}\right).$

Existence of real roots

Each polynomial in an orthogonal sequence has all n of its roots real, distinct, and strictly inside the interval of orthogonality. (proof)

(Anyone who has graphed polynomials in high school knows that it is very rare for a randomly-chosen high-degree polynomial to have all of its roots real.)

Interlacing of roots

The roots of each polynomial lie strictly between the roots of the next higher polynomial in the sequence. (proof)

Differential equations leading to orthogonal polynomials

A very important class of orthogonal polynomials arises 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 a eigenvector/eigenvalue problems: Letting D be the differential operator, $D(f)=Qf''+Lf'\,$, 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 ${\lambda }_{0},{\lambda }_{1},{\lambda }_{2},\dots \,$ 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, P_n(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. 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 e_n depend on the standardization. The standard values of e_n will be given in the tables below.

The numbers λ_n

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 r^th derivative of $P_{n}$. (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)

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

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

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

and reversing the subscripts yields

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

subtracting and integrating:

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

but it can be seen that

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

so that:

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

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

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

for $m\neq n$.

The classical orthogonal polynomials

The class of polynomials arising from the differential equation described above have many important applications in such areas as mathematical physics, interpolation theory, the theory of random matrices, computer approximations, and many others. All of these polynomial sequences are equivalent, under scaling and/or shifting of the domain, and standardizing of the polynomials, to more restricted classes. Those restricted classes are the "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 polynomials

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 $\alpha$ and $\beta$ 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 $\alpha$ and $\beta$ are required to be greater than −1. (This puts the root of L inside the interval of orthogonality.)

When $\alpha$ and $\beta$ 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 $\alpha$ and $\beta$ 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 $\alpha$ 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 $\alpha$ 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

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

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 $0{\leq }m{\leq }\ell$, 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 m^th 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

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

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,

$T_{n}(x)=\cos(n\,\cos ^{-1}(x)).$

(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 $\alpha$, 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 $\alpha =0$ version of these:

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

The differential equation is

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

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 $\alpha$ 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

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

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

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

is a solution of

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

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.

Some 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.

Constructing orthogonal polynomials by using moments

Let

$\mu _{n}=\int _{\mathbb {R} }x^{n}\,d\mu$

be the moments of a measure μ. Then the polynomial sequence defined by

$p_{n}(x)=\det \left[{\begin{matrix}\mu _{0}&\mu _{1}&\mu _{2}&\cdots &\mu _{n}\\\mu _{1}&\mu _{2}&\mu _{3}&\cdots &\mu _{n+1}\\\mu _{2}&\mu _{3}&\mu _{4}&\cdots &\mu _{n+2}\\\vdots &\vdots &\vdots &&\vdots \\\mu _{n-1}&\mu _{n}&\mu _{n+1}&\cdots &\mu _{2n-1}\\1&x&x^{2}&\cdots &x^{n}\end{matrix}}\right]$

is a sequence of orthogonal polynomials with respect to the measure μ. To see this, consider the inner product of p_n(x) with x^k for any k < n. We will see that the value of this inner product is zero.

${\begin{aligned}\int _{\mathbb {R} }x^{k}p_{n}(x)\,d\mu &{}=\int _{\mathbb {R} }x^{k}\det \left[{\begin{matrix}\mu _{0}&\mu _{1}&\mu _{2}&\cdots &\mu _{n}\\\mu _{1}&\mu _{2}&\mu _{3}&\cdots &\mu _{n+1}\\\mu _{2}&\mu _{3}&\mu _{4}&\cdots &\mu _{n+2}\\\vdots &\vdots &\vdots &&\vdots \\\mu _{n-1}&\mu _{n}&\mu _{n+1}&\cdots &\mu _{2n-1}\\1&x&x^{2}&\cdots &x^{n}\end{matrix}}\right]\,d\mu \\\\&{}=\int _{\mathbb {R} }\det \left[{\begin{matrix}\mu _{0}&\mu _{1}&\mu _{2}&\cdots &\mu _{n}\\\mu _{1}&\mu _{2}&\mu _{3}&\cdots &\mu _{n+1}\\\mu _{2}&\mu _{3}&\mu _{4}&\cdots &\mu _{n+2}\\\vdots &\vdots &\vdots &&\vdots \\\mu _{n-1}&\mu _{n}&\mu _{n+1}&\cdots &\mu _{2n-1}\\x^{k}&x^{k+1}&x^{k+2}&\cdots &x^{k+n}\end{matrix}}\right]\,d\mu \\\\&{}=\det \left[{\begin{matrix}\mu _{0}&\mu _{1}&\mu _{2}&\cdots &\mu _{n}\\\mu _{1}&\mu _{2}&\mu _{3}&\cdots &\mu _{n+1}\\\mu _{2}&\mu _{3}&\mu _{4}&\cdots &\mu _{n+2}\\\vdots &\vdots &\vdots &&\vdots \\\mu _{n-1}&\mu _{n}&\mu _{n+1}&\cdots &\mu _{2n-1}\\\displaystyle \int _{\mathbb {R} }x^{k}\,d\mu &\displaystyle \int _{\mathbb {R} }x^{k+1}\,d\mu &\displaystyle \int _{\mathbb {R} }x^{k+2}\,d\mu &\cdots &\displaystyle \int _{\mathbb {R} }x^{k+n}\,d\mu \end{matrix}}\right]\\\\&{}=\det \left[{\begin{matrix}\mu _{0}&\mu _{1}&\mu _{2}&\cdots &\mu _{n}\\\mu _{1}&\mu _{2}&\mu _{3}&\cdots &\mu _{n+1}\\\mu _{2}&\mu _{3}&\mu _{4}&\cdots &\mu _{n+2}\\\vdots &\vdots &\vdots &&\vdots \\\mu _{n-1}&\mu _{n}&\mu _{n+1}&\cdots &\mu _{2n-1}\\\mu _{k}&\mu _{k+1}&\mu _{k+2}&\cdots &\mu _{k+n}\end{matrix}}\right]\\\\&{}=0{\text{ if }}k<n,{\text{ since the matrix has two identical rows}}.\end{aligned}}$

Thus p_n(x) is orthogonal to x^k for all k < n. That means this is a sequence of orthogonal polynomials for the measure μ.

Table of classical orthogonal polynomials

Name, and conventional symbol	Chebyshev, $\ T_{n}$	Chebyshev (second kind), $\ U_{n}$	Legendre, $\ P_{n}$	Hermite, $\ H_{n}$
Limits of orthogonality	$-1,1\,$	$-1,1\,$	$-1,1\,$	$-\infty ,\infty$
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, $h_{n}\,$	$\left\{{\begin{matrix}\pi &:~n=0\\\pi /2&:~n\neq 0\end{matrix}}\right.$	$\pi /2\,$	${\frac {2}{2n+1}}$	$2^{n}\,n!\,{\sqrt {\pi }}$
Leading term, $k_{n}\,$	$2^{n-1}\,$	$2^{n}\,$	${\frac {(2n)!}{2^{n}\,(n!)^{2}}}\,$	$2^{n}\,$
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 $\alpha \neq 0$	$P_{n}^{(\alpha ,\beta )}(1)={\frac {\Gamma (n+1+\alpha )}{n!\,\Gamma (1+\alpha )}}\,$
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, $c_{n}\,$	${\frac {n+2{\alpha }-1}{n+1}}\,$	${\frac {(n+\alpha )(n+\beta )(2n+2+\alpha +\beta )}{(n+1)(n+1+\alpha +\beta )(2n+\alpha +\beta )}}$

See also

• Polynomial sequences of binomial type
• Generalized Fourier series
• Sheffer sequence
• Appell sequence
• Umbral calculus
• Secondary measure

References

• 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.
• Gabor Szego (1939). Orthogonal Polynomials. Colloquium Publications - American Mathematical Society. ISBN 0-8218-1023-5.
• Dunham Jackson (1941, 2004). Fourier Series and Orthogonal Polynomials. New York: Dover. ISBN 0-486-43808-2. {{cite book}}: Check date values in: |year= (help)
• Refaat El Attar (2006). Special Functions and Orthogonal Polynomials. Lulu Press, Morrisville NC 27560. ISBN 1-4116-6690-9.
• Theodore Seio Chihara (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. ISBN 0-677-04150-0.

Further reading

• Ismail, Mourad E. H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable. ISBN 0-521-78201-5.
• Vilmos Totik (2005). "Orthogonal Polynomials". Surveys in Approximation Theory. 1: 70–125.