In mathematics, an orthogonal polynomial sequence is an infinite sequence of polynomials p₀(x), p₁(x), p₂(x) ... , in which each p_n(x) has degree n, and such that any two different polynomials in the sequence are orthogonal to each other in the following sense: Euclid, detail from The School of Athens by Raphael. ... In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

One can define an inner product on functions, (analogous to the ordinary "dot product" for vectors), by integrating the product of the functions:

More generally, one can put a fixed "weight function" W(x) into the integral:

Two functions are orthogonal to each other if their inner product is zero, in the same way that ordinary vectors are orthogonal (perpendicular) if their dot product is zero.

Such an inner product makes the set of all functions of finite norm a Hilbert space.

So a polynomial sequence is an orthogonal sequence with respect to the weight function W when any two different polynomials in the sequence are orthogonal, using that weight function, i.e., // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity α resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ...

The interval of integration is called the interval of orthogonality. It might be infinite at one or both ends.

The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Stieltjes. It evolved into a field rich in applications to many areas of mathematics and physics. Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... Thomas Joannes Stieltjes (December 29, 1856 – December 31, 1894) was a Dutch mathematician. ... A Superconductor demonstrating the Meissner Effect. ...

The simplest orthogonal polynomials are the Legendre polynomials, for which the interval of orthogonality is [−1, 1] and the weight function is simply 1: In mathematics, Legendre functions are solutions to Legendres differential equation: They are named after Adrien-Marie Legendre. ...

These are all orthogonal over [−1, 1]:

We require that the weight function be strictly positive in the interior of the interval of orthogonality. In some cases, it may be zero, or go off to infinity, at the end points. The integral of the weight function times any polynomial must be finite.

Now any sequence of polynomials , with each having degree k, is a basis for the (infinite-dimensional) vector space of all polynomials. An orthogonal sequence is just a sequence that comprises an orthogonal basis for that space, relative to the given inner product.

The Gram-Schmidt process can turn any basis for a vector space into an orthogonal basis, by starting with one vector and then repeatedly incorporating new vectors while making each new vector orthogonal to all the previous ones. This is done by subtracting suitable linear combinations of the previous vectors. Doing this for polynomials is often used as an exercise in elementary linear algebra courses. It results in the Legendre polynomials. In mathematics and numerical analysis, the Gram-Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ...

When making an orthogonal basis, one may be tempted to make an orthonormal basis, that is, one in which . 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.

Once a polynomial sequence has been standardized, we can define the norm. Let In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

The norm is the square root of this. The values of for the standardized classical polynomials will be listed in the table below. Using , we have

where δ_mn is the Kronecker delta. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

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 , any n^th-degree polynomial can be expanded in terms of . That is, there are coefficients such that

Proof by mathematical induction. Choose so that the term of matches that of . Then is an n − 1^st degree polynomial. Continue downward. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. ...

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 . is orthogonal to each of them.

Recurrence relations

Any orthogonal sequence has a recurrence formula relating any three consecutive polynomials in the sequence. In mathematics, a recurrence relation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

The coefficients a, b, and c depend on n. They also depend on the standardization, obviously. (proof) // Proof of the recurrence relation Any orthogonal series has a recurrence formula relating any three consecutive polynomials in the series. ...

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:

and h_j be the inner product of p_j with itself:

we have

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) // Proof of the recurrence relation Any orthogonal series has a recurrence formula relating any three consecutive polynomials in the series. ...

(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) // Proof of the recurrence relation Any orthogonal series has a recurrence formula relating any three consecutive polynomials in the series. ...

Differential equations leading to orthogonal polynomials

A very important class of orthogonal polynomials arises from a differential equation of the form

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 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. In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form where the functions p(x), q(x), and w(x) are specified at the outset... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

The solutions of this differential equation have singularities unless λ takes on specific values. There is a series of numbers that lead to a series of polynomial solutions 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 , each having degree n, and corresponding to a number .
• The interval of orthogonality is bounded by whatever roots Q has.
• The root of L is inside the interval of orthogonality.
• Letting , the polynomials are orthogonal under the weight function
• 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

This is known as Rodrigues formula. It is often written

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

(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 .

Then

Now multiply the differential equation

by R/Q, getting

or

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

Third form for the differential equation

Let .

Then

Now multiply the differential equation

by S/Q, getting

or

But , so

or, letting u = Sy,

Formulas involving derivatives

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

• (orthogonality) For fixed r, the polynomial sequence are orthogonal, weighted by .
• (generalized Rodriguez formula) is proportional to .
• (differential equation) is a solution of , where is the same function as , that is,
• (differential equation, second form) is a solution of

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.

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:

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". In probability theory and statistics, a random matrix is a matrix-valued random variable. ...

• 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 . 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 , and has Q = x. They can then be standardized into the Associated Laguerre polynomials . The plain Laguerre polynomials 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 , and has Q = 1. They can then be standardized into the Hermite polynomials .

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 α and β in the Jacobi polynomials, written . We have and . Both α and β are required to be greater than −1. (This puts the root of L inside the interval of orthogonality.)

When α and β are not equal, these polynomials are not symmetrical about x=0.

The differential equation

is Jacobi's equation.

For further details, see Jacobi polynomials. In mathematics, Jacobi polynomials are a class of orthogonal polynomials. ...

Gegenbauer polynomials

When one sets the parameters α and β in the Jacobi polynomials equal to each other, one obtains the Gegenbauer or ultraspherical polynomials. They are written , and defined as

We have and . is required to be greater than −1/2.

(Incidentally, the standardization given in the table below would make no sense for α=0 and nǂ0, because it would set the polynomials to zero. In that case, the accepted standardization sets instead of the value given in the table.)

Ignoring the above considerations, the parameter α is closely related to the derivatives of :

or, more generally:

All the other classical Jacobi-like polynomials (Legendre, etc.) are special cases of the Gegenbauer polynomials, obtained by choosing a value of α and choosing a standardization.

For further details, see Gegenbauer polynomials. In mathematics, Gegenbauer polynomials or ultraspherical polynomials are a class of orthogonal polynomials. ...

Legendre polynomials

The differential equation is

This is Legendre's equation.

The second form of the differential equation is

The recurrence relation is

Rodrigues formula is

Associated Legendre functions

The Associated Legendre functions, denoted where l and m are integers with , are defined as In mathematics, the associated Legendre polynomials, named after Adrien-Marie Legendre, are defined by: These differ from the Legendre polynomials. ...

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.

(The associated Legendre functions are not polynomials when m is odd.)

They have a recurrence relation:

For fixed m, the sequence are orthogonal over [−1, 1], with weight 1.

For given m, are the solutions of

These functions are most useful when the argument is reparameterized in terms of angles:

For fixed m, are orthogonal, parameterized by θ over [0,π], with weight sinθ:

In terms of θ, are solutions of

More precisely, given an integer m0, the above equation has nonsingular solutions only when for an integer, and the solutions are proportional to .

Spherical harmonics

What makes these functions useful is that they are central to the solution of the equation on the surface of a sphere. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

When the partial differential equation In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

is solved by the method of separation of variables, one gets a φ-dependent part sin(mφ) or cos(mφ) for integer m≥0, and an equation for the θ-dependent part In mathematics, separation of variables is any of several methods of solving ordinary and partial differential equations. ...

for which the solutions are with and .

Therefore, the equation

has nonsingular separated solutions only when , and the solutions are proportional to

and

For each choice of , there are functions for the various values of m and choices of sine and cosine. They are all orthogonal in both l and m when integrated over the surface of the sphere.

They are usually written in terms of complex exponentials:

The functions are called spherical harmonics. The quantity in the square root is a normalizing factor. These functions form a complete orthonormal set of functions in the sense of Fourier series. In mathematics, the spherical harmonics are an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ... Fourier series are a mathematical technique for analyzing an arbitrary periodic function by decomposing the function into a sum of much simpler sinusoidal component functions, which differ from each other only in amplitude and frequency. ...

When a 3-dimensional spherically symmetric partial differential equation is solved by the method of separation of variables in spherical coordinates, the part that remains after removal of the radial part is typically of the form , and hence the solutions are spherical harmonics.

For further details, see Legendre polynomials. In mathematics, Legendre functions are solutions to Legendres differential equation: They are named after Adrien-Marie Legendre. ...

Chebyshev polynomials

The differential equation is

This is Chebyshev's equation.

The recurrence relation is

Rodrigues formula is

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

(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

We have:

For further details, including the expressions for the first few polynomials, see Chebyshev polynomials. In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are a sequence of orthogonal polynomials which are related to de Moivres formula and which are easily defined recursively, like Fibonacci or Lucas numbers. ...

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 . There is a parameter α, 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 α = 0 version of these:

The differential equation is

This is Laguerre's equation.

The second form of the differential equation is

The recurrence relation is

Rodrigues formula is

The parameter α is closely related to the derivatives of :

or, more generally:

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

is a solution of

This can be further manipulated. When is an integer, and :

is a solution of

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

This equation arises in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom. In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ...

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. In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are a polynomial sequence defined by These polynomials are orthogonal to each other with respect to the inner product given by Also, for each n, Ln(x) is a solution of Laguerres equation which is a second-order...

Hermite polynomials

The differential equation is

This is Hermite's equation.

The second form of the differential equation is

The third form is

The recurrence relation is

Rodrigues formula is

The first few Hermite polynomials are

One can define the associated Hermite functions

Because the multiplier is proportional to the square root of the weight function, these functions are orthogonal over with no weight function.

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

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 Fourier transform. The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...

Some authors, particularly probabilists, use an alternate definition of the Hermite polynomials, with a weight function of instead of . This is generally named with the two-letter symbol . It could be defined as

For further details, see Hermite polynomials. In mathematics, the Hermite polynomials, named in honor of Charles Hermite (Hermite is pronounced air MEET), are a polynomial sequence defined either by (the probabilists Hermite polynomials), or sometimes by (the physicists Hermite polynomials). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. ...

Table of classical orthogonal polynomials

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are a sequence of orthogonal polynomials which are related to de Moivres formula and which are easily defined recursively, like Fibonacci or Lucas numbers. ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are a sequence of orthogonal polynomials which are related to de Moivres formula and which are easily defined recursively, like Fibonacci or Lucas numbers. ... In mathematics, Legendre functions are solutions to Legendres differential equation: They are named after Adrien-Marie Legendre. ... In mathematics, the Hermite polynomials, named in honor of Charles Hermite (Hermite is pronounced air MEET), are a polynomial sequence defined either by (the probabilists Hermite polynomials), or sometimes by (the physicists Hermite polynomials). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. ... In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are a polynomial sequence defined by These polynomials are orthogonal to each other with respect to the inner product given by Also, for each n, Ln(x) is a solution of Laguerres equation which is a second-order... In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are a polynomial sequence defined by These polynomials are orthogonal to each other with respect to the inner product given by Also, for each n, Ln(x) is a solution of Laguerres equation which is a second-order... In mathematics, Gegenbauer polynomials or ultraspherical polynomials are a class of orthogonal polynomials. ... In mathematics, Jacobi polynomials are a class of orthogonal polynomials. ...

See also

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

Definition In mathematics, a polynomial sequence, i. ... In mathematical analysis, there are many potentially useful generalizations of Fourier series. ... In mathematics, a polynomial sequence, i. ... In mathematics, before the 1970s, the term umbral calculus was understood to mean the surprising similarities between otherwise unrelated polynomial equations, and certain shadowy techniques that can be used to prove them. ...

References

• Milton Abramowitz and Irene A. Stegun, eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York:Dover. ISBN 0486612724. (See chapter 22)
• Gabor Szego (1939). Orthogonal Polynomials, Colloquium Publications - American Mathematical Society. ISBN 0821810235.
• Dunham Jackson (1941, 2004). Fourier Series and Orthogonal Polynomials, New York:Dover. ISBN 0486438082.
• Refaat El Attar (2006). Special Functions and Orthogonal Polynomials, Lulu Press, Morrisville NC 27560. ISBN 1411666909.