Bateman polynomials

2013-11-21T19:52:33Z

76.98.128.9:

In [[mathematics]], an '''orthogonal polynomial sequence''' is a family of [[polynomial]]s
such that any two different polynomials in the sequence are [[orthogonality|orthogonal]] to each other under some [[inner product]].

The most widely used orthogonal polynomials are the [[classical orthogonal polynomials]], consisting of the [[Hermite polynomials]], the [[Laguerre polynomials]], the [[Jacobi polynomials]] together with their special cases the [[Gegenbauer polynomials]], the [[Chebyshev polynomials]], and the [[Legendre polynomials]].

The field of orthogonal polynomials developed in the late 19th century from a study of [[continued fraction]]s by [[Pafnuty Chebyshev|P. L. Chebyshev]] and was pursued by [[Andrey Markov|A.A. Markov]] and [[Thomas Joannes Stieltjes|T.J. Stieltjes]]. Some of the mathematicians who have worked on orthogonal polynomials include [[Gábor Szegő]], [[Sergei Natanovich Bernstein|Sergei Bernstein]], [[Naum Akhiezer]], [[Arthur Erdélyi]], [[Yakov Geronimus]], [[Wolfgang Hahn]], [[Theodore Seio Chihara]], [[Mourad Ismail]], [[Waleed Al-Salam]], and [[Richard Askey]].

== Definition for 1-variable case for a real measure ==

Given any non-decreasing function ''α'' on the real numbers, we can define the [[Lebesgue–Stieltjes integral]]
:<math>\int f(x)d\alpha(x)</math>
of a function ''f''. If this integral is finite for all polynomials ''f'', we can
define an inner product on pairs of polynomials ''f'' and ''g'' by

:<math>\langle f, g \rangle = \int f(x) g(x) \; d\alpha(x).</math>

This operation is a positive semidefinite [[inner product space|inner product]] on the [[vector space]] of all polynomials, and is positive definite if the function α has an infinite number of points of growth. It induces a notion of orthogonality in the usual way, namely that two polynomials are orthogonal if their inner product is zero.

Then the sequence (''P''''n'')''n''=0∞ of orthogonal polynomials is defined by the relations

:<math> \deg P_n = n~, \quad \langle P_m, \, P_n \rangle = 0 \quad \text{for} \quad m \neq n~.</math>

In other words, the sequence is obtained from the sequence of monomials 1, ''x'', ''x''2, ... by the [[Gram–Schmidt process]] with respect to this inner product.

Usually the sequence is required to be [[orthonormal]], namely,

:<math> \langle P_n, P_n \rangle = 1~, </math>

however, other normalisations are sometimes used.

===Absolutely continuous case===

Sometimes we have
:<math>\displaystyle d\alpha(x) = W(x)dx</math>
where
:<math>W : [x_1, x_2] \to \mathbb{R}</math>
is a non-negative function with support on some interval [''x''1, ''x''2] in the real line (where ''x''1 = −∞ and ''x''2 = ∞ are allowed). Such a ''W'' is called a '''weight function'''.
Then the inner product is given by
:<math>\langle f, g \rangle = \int_{x_1}^{x_2} f(x) g(x) W(x) \; dx.</math>
However there are many examples of orthogonal polynomials where the measure dα(''x'') has points with non-zero measure where the function α is discontinuous, so cannot be given by a weight function ''W'' as above.

==Examples of orthogonal polynomials==

The most commonly used orthogonal polynomials are orthogonal for a measure with support in a real interval. This includes:
*The classical orthogonal polynomials ([[Jacobi polynomials]], [[Laguerre polynomials]], [[Hermite polynomials]], and their special cases [[Gegenbauer polynomials]], [[Chebyshev polynomials]] and [[Legendre polynomials]]).
*The [[Wilson polynomials]], which generalize the Jacobi polynomials. They include many orthogonal polynomials as special cases, such as the [[Meixner–Pollaczek polynomials]], the [[continuous Hahn polynomials]], the [[continuous dual Hahn polynomials]], and the classical polynomials, described by the [[Askey scheme]]
*The [[Askey–Wilson polynomials]] introduce an extra parameter ''q'' into the Wilson polynomials.

[[Discrete orthogonal polynomials]] are orthogonal with respect to some discrete measure. Sometimes the measure has finite support, in which case the family of orthogonal polynomials is finite, rather than an infinite sequence. The [[Racah polynomials]] are examples of discrete orthogonal polynomials, and include as special cases the [[Hahn polynomials]] and [[dual Hahn polynomials]], which in turn include as special cases the [[Meixner polynomials]], [[Krawtchouk polynomials]], and [[Charlier polynomials]].

[[Sieved orthogonal polynomials]], such as the [[sieved ultraspherical polynomials]], [[sieved Jacobi polynomials]], and [[sieved Pollaczek polynomials]], have modified recurrence relations.

One can also consider orthogonal polynomials for some curve in the complex plane. The most important case (other than real intervals) is when the curve is the unit circle, giving [[orthogonal polynomials on the unit circle]], such as the [[Rogers–Szegő polynomials]].

There are some families of orthogonal polynomials that are orthogonal on plane regions such as triangles or disks. They can sometimes be written in terms of Jacobi polynomials. For example, [[Zernike polynomials]] are orthogonal on the unit disk.

==Properties==

Orthogonal polynomials of one variable defined by a non-negative measure on the real line have the following properties.

===Relation to moments===

The orthogonal polynomials ''P''''n'' can be expressed in terms of the [[moment (mathematics)|moments]]

:<math> m_n = \int x^n d\alpha(x) </math>

as follows:

:<math> P_n(x) = c_n \, \det \begin{bmatrix}
m_0 & m_1 & m_2 &\cdots & m_n \\
m_1 & m_2 & m_3 &\cdots & m_{n+1} \\
&&\cdots&& \\
m_{n-1} &m_n& m_{n+1} &\cdots &m_{2n-1}\\
1 & x & x^2 & \cdots & x^{n}
\end{bmatrix}~,</math>

where the constants ''c''''n'' are arbitrary (depend on the normalisation of ''P''''n'').

===Recurrence relation===

The polynomials ''P''''n'' satisfy a recurrence relation of the form

:<math> P_n(x) = (A_n x + B_n) P_{n-1}(x) + C_n P_{n-2}(x)~.</math>

See [[Favard's theorem]] for a converse result.

===Christoffel–Darboux formula===

{{main|Christoffel–Darboux formula}}

===Zeros===

If the measure d''α'' is supported on an interval [''a'', ''b''], all the zeros of ''P''''n'' lie in [''a'', ''b'']. Moreover, the zeros have the following interlacing property: if ''m''>''n'', there is a zero of ''P''''m'' between any two zeros of ''P''''n''.

==Multivariate orthogonal polynomials==

The [[Macdonald polynomials]] are orthogonal polynomials in several variables, depending on the choice of an affine root system. They include many other families of multivariable orthogonal polynomials as special cases, including the [[Jack polynomials]], the [[Hall–Littlewood polynomials]], the [[Heckman–Opdam polynomials]], and the [[Koornwinder polynomials]]. The [[Askey–Wilson polynomials]] are the special case of Macdonald polynomials for a certain non-reduced root system of rank 1.

== See also ==

* [[Appell sequence]]
* [[Askey scheme]] of hypergeometric orthogonal polynomials
* [[Binomial type|Polynomial sequences of binomial type]]
* [[Biorthogonal polynomials]]
* [[Generalized Fourier series]]
* [[Secondary measure]]
* [[Sheffer sequence]]
* [[Umbral calculus]]

== References ==

* {{Abramowitz_Stegun_ref|22|773}}
* {{cite book | first=Theodore Seio|last= Chihara | title= An Introduction to Orthogonal Polynomials | publisher= Gordon and Breach, New York | year=1978 | isbn = 0-677-04150-0}}
*{{Cite journal | last1=Chihara | first1=Theodore Seio | title=Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999) | doi=10.1016/S0377-0427(00)00632-4 | mr=1858267 | year=2001 | journal=Journal of Computational and Applied Mathematics | issn=0377-0427 | volume=133 | issue=1 | chapter=45 years of orthogonal polynomials: a view from the wings | pages=13–21 | postscript={{inconsistent citations}}}}
*{{Cite journal | last1=Foncannon | first1=J. J. | title=Review of ''Classical and quantum orthogonal polynomials in one variable'' by Mourad Ismail | publisher=Springer New York | doi=10.1007/BF02985757 | year=2008 | journal=[[The Mathematical Intelligencer]] | issn=0343-6993 | volume=30 | pages=54–60 | last2=Foncannon | first2=J. J. | last3=Pekonen | first3=Osmo | postscript={{inconsistent citations}}}}
*{{cite book | last=Ismail|first=Mourad E. H. | title=Classical and Quantum Orthogonal Polynomials in One Variable | year=2005 | isbn=0-521-78201-5 | url = http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521782012 | publisher=Cambridge Univ. Press | location=Cambridge}}
* {{cite book | first=Dunham |last=Jackson | title= Fourier Series and Orthogonal Polynomials | location= New York | publisher=Dover | origyear=1941|year= 2004 | isbn = 0-486-43808-2}}
*{{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
*{{springer|title=Orthogonal polynomials|id=p/o070340}}
*{{Cite book | last1=Szegő | first1=Gábor | title=Orthogonal Polynomials | url=http://books.google.com/books?id=3hcW8HBh7gsC | publisher= American Mathematical Society | series=Colloquium Publications | isbn=978-0-8218-1023-1 | mr=0372517 | year=1939 | volume=XXIII | postscript={{inconsistent citations}}}}
*{{cite journal | first= Vilmos|last= Totik | authorlink = Vilmos Totik | year = 2005 | title = Orthogonal Polynomials | journal = Surveys in Approximation Theory | volume = 1 | pages = 70–125 | arxiv = math.CA/0512424}}

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[[Category:Orthogonal polynomials| ]]

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Bateman polynomials