For given polynomials$Q,L:\mathbb {R} \to \mathbb {R}$ and $\forall \,n\in \mathbb {N} _{0}$ the classical orthogonal polynomials $f_{n}:\mathbb {R} \to \mathbb {R}$ are characterized by being solutions of the differential equation
with to be determined constants $\lambda _{n}\in \mathbb {R}$.
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.
Definition
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In general, the orthogonal polynomials $P_{n}$ with respect to a weight $W:\mathbb {R} \rightarrow \mathbb {R} ^{+}$ satisfy
The Jacobi polynomials with $\alpha =\beta$ are called the Gegenbauer polynomials (with parameter $\gamma =\alpha +1/2$)
For $\alpha =\beta =0$, these are called the Legendre polynomials (for which the interval of orthogonality is [−1, 1] and the weight function is simply 1):
The classical orthogonal polynomials arise 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 as 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 λ_{0}, λ_{1}, λ_{2}, ... that led to a series of polynomial solutions P_{0}, P_{1}, P_{2}, ... if one of the following sets of conditions are met:
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.
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.
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}, ..., 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 $R(x)=e^{\int {\frac {L(x)}{Q(x)}}\,dx}$, 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
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Under the assumptions of the preceding section,
P_{n}(x) is proportional to ${\frac {1}{W(x)}}\ {\frac {d^{n}}{dx^{n}}}\left(W(x)[Q(x)]^{n}\right).$
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 $WQ^{r}$.
(generalized Rodrigues' formula) P^{[r]} _{n} 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^{[r]} _{n} is a solution of ${Q}\,y''+(rQ'+L)\,y'+[\lambda _{n}-\lambda _{r}]\,y=0$, where λ_{r} is the same function as λ_{n}, that is, $\lambda _{r}=-r\left({\frac {r-1}{2}}Q''+L'\right)$
(differential equation, second form) P^{[r]} _{n} 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.
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:
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:
$\int _{a}^{b}{\frac {R}{Q}}f_{m}f_{n}\,dx=0$
for $m\neq n$.
Derivation from differential equation
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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 − x^{2}. 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 polynomial
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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.
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
We have $Q(x)=1-x^{2}$ and
$L(x)=-(2\alpha +1)\,x$.
The parameter $\alpha$ 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 $C_{n}^{(\alpha )}$:
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.
These polynomials have the property that, in the interval of orthogonality,
$T_{n}(x)=\cos(n\,\arccos(x)).$
(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 $U_{n}$
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
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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
$x\,y''+(\alpha +1-x)\,y'+\lambda \,y=0{\text{ with }}\lambda =n.$
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.
Many 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
Characterizations of classical orthogonal polynomials
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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
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The following table summarises the properties of the classical orthogonal polynomials.^{[3]}
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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.