Classical orthogonal polynomials

Differential equation

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

${\displaystyle Q(x)f^{″}+L(x)f^{\prime }+\lambda f=0}$

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, ${\displaystyle D(f)=Q{f}^{″}+L{f}^{\prime }}$, 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).

Rodrigues' formula

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

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

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

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

The numbers λn

Under the assumptions of the preceding section, we have

${\displaystyle {\lambda }_n=-n\left(\frac{n-1}{2}{Q}^{″}+L^{\prime }\right).}$

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

Second form for the differential equation

Let

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

Then

${\displaystyle (R{y}^{\prime }{)}^{\prime }=R{y}^{″}+R^{\prime }{y}^{\prime }=R{y}^{″}+\frac{RL}{Q}{y}^{\prime }.}$

Now multiply the differential equation

${\displaystyle Q{y}^{″}+L{y}^{\prime }+\lambda y=0}$

by R/Q, getting

${\displaystyle R{y}^{″}+\frac{RL}{Q}{y}^{\prime }+\frac{R\lambda }{Q}y=0}$

or

${\displaystyle (R{y}^{\prime }{)}^{\prime }+\frac{R\lambda }{Q}y=0.}$

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

Third form for the differential equation

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

Then

${\displaystyle S^{\prime }=\frac{SL}{2Q}.}$

Now multiply the differential equation

${\displaystyle Q{y}^{″}+L{y}^{\prime }+\lambda y=0}$

by S/Q, getting

${\displaystyle S{y}^{″}+\frac{SL}{Q}{y}^{\prime }+\frac{S\lambda }{Q}y=0}$

or

${\displaystyle S{y}^{″}+2S^{\prime }{y}^{\prime }+\frac{S\lambda }{Q}y=0}$

But ${\displaystyle (Sy{)}^{″}=S{y}^{″}+2S^{\prime }{y}^{\prime }+S^{″}y}$, so

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

or, letting u = Sy,

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

Formulas involving derivatives

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 ${\displaystyle W{Q}^r}$.
• (generalized Rodrigues' formula) P^[r]
  _n is proportional to ${\displaystyle \frac{1}{W(x)[Q(x){]}^r}\frac{d^{n-r}}{d{x}^{n-r}}\left(W(x)[Q(x){]}^n\right).}$
• (differential equation) P^[r]
  _n is a solution of ${\displaystyle Q{y}^{″}+(r{Q}^{\prime }+L)y^{\prime }+[{\lambda }_n-{\lambda }_r]y=0}$, where λ_r is the same function as λ_n, that is, ${\displaystyle {\lambda }_r=-r\left(\frac{r-1}{2}{Q}^{″}+L^{\prime }\right)}$
• (differential equation, second form) P^[r]
  _n is a solution of ${\displaystyle (R{Q}^r{y}^{\prime }{)}^{\prime }+[{\lambda }_n-{\lambda }_r]R{Q}^{r-1}y=0}$

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.

• ${\displaystyle P_n^{[r]}=a{P}_{n+1}^{[r+1]}+b{P}_n^{[r+1]}+c{P}_{n-1}^{[r+1]}}$
• ${\displaystyle P_n^{[r]}=(ax+b)P_n^{[r+1]}+c{P}_{n-1}^{[r+1]}}$
• ${\displaystyle Q{P}_n^{[r+1]}=(ax+b)P_n^{[r]}+c{P}_{n-1}^{[r]}}$

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:

• ${\displaystyle 2T_m(x)T_n(x)=T_{m+n}(x)+T_{m-n}(x)}$
• ${\displaystyle H_{2n}(x)=(-4{)}^{n}n!L_n^{(-1/2)}(x^2)}$
• ${\displaystyle H_{2n+1}(x)=2(-4{)}^{n}n!x{L}_n^{(1/2)}(x^2)}$

Orthogonality

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

${\displaystyle R{f}_m{\ddot{f}}_n+\frac{R}{Q}L{f}_m{\dot{f}}_n+\frac{R}{Q}{\lambda }_n{f}_m{f}_n=0}$

and reversing the subscripts yields

${\displaystyle R{f}_n{\ddot{f}}_m+\frac{R}{Q}L{f}_n{\dot{f}}_m+\frac{R}{Q}{\lambda }_m{f}_n{f}_m=0}$

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 ${\displaystyle {\lambda }_m\ne {\lambda }_n}$ for ${\displaystyle m\ne n}$, then the orthogonality relationship will hold:

${\displaystyle {\int }_a^b\frac{R}{Q}{f}_m{f}_{n}dx=0}$

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

Derivation from differential equation

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

• Every Jacobi-like polynomial sequence can have its domain shifted and/or scaled so that its interval of orthogonality is [−1, 1], and has Q = 1 − x². They can then be standardized into the Jacobi polynomials ${\displaystyle P_n^{(\alpha ,\beta )}}$. 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 ${\displaystyle [0,\mathrm{\infty })}$, 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 ${\displaystyle (-\mathrm{\infty },\mathrm{\infty })}$, 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.

Jacobi polynomial

The Jacobi-like polynomials, once they have had their domain shifted and scaled so that the interval of orthogonality is [−1, 1], still have two parameters to be determined. They are ${\displaystyle \alpha }$ 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

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

is Jacobi's equation.

For further details, see Jacobi polynomials.

Gegenbauer 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

${\displaystyle C_n^{(\alpha )}(x)=\frac{\mathrm{\Gamma }(2\alpha +n)\mathrm{\Gamma }(\alpha +1/2)}{\mathrm{\Gamma }(2\alpha )\mathrm{\Gamma }(\alpha +n+1/2)}{P}_n^{(\alpha -1/2,\alpha -1/2)}(x).}$

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:

${\displaystyle C_n^{(\alpha +m)}(x)=\frac{\mathrm{\Gamma }(\alpha )}{2^m\mathrm{\Gamma }(\alpha +m)}{C}_{n+m}^{(\alpha )[m]}(x).}$

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.

Legendre polynomials

The differential equation is

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

This is Legendre's equation.

The second form of the differential equation is:

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

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

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

For further details, see Legendre polynomials.

Associated Legendre polynomials

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

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

Chebyshev polynomials

The differential equation is

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

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

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

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:

${\displaystyle U_n=\frac{1}{n+1}{T}_{n+1}^{\prime }.}$

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 ${\displaystyle L_n^{(\alpha )}}$. 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

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

This is Laguerre's equation.

The second form of the differential equation is

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

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

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

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

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

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

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

is a solution of

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

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.

Hermite polynomials

The differential equation is

${\displaystyle y^{″}-2x{y}^{\prime }+\lambda y=0,\text{with}\lambda =2n.}$

This is Hermite's equation.

The second form of the differential equation is

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

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

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

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 ${\displaystyle (-\mathrm{\infty },\mathrm{\infty })}$ 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

${\displaystyle H{e}_n(x)=2^{-n/2}{H}_n\left(\frac{x}{\sqrt{2}}\right).}$

For further details, see Hermite polynomials.

Characterizations of classical orthogonal polynomials

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

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

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

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

Table of classical orthogonal polynomials

The following table summarises the properties of the classical orthogonal polynomials.