Classical orthogonal polynomials

The general term for Jacobi polynomials; Hermite polynomials; and Laguerre polynomials. These systems of orthogonal polynomials have the following properties in common:

1) The weight function $ \phi ( x) $ on the interval of orthogonality $ ( a , b ) $ satisfies the Pearson differential equation

$$ \frac{\phi ^ \prime ( x) }{\phi ( x) } = \ \frac{p _ {0} + p _ {1} x }{q _ {0} + q _ {1} x + q _ {2} x ^ {2} } \equiv \ \frac{A ( x) }{B ( x) } ,\ \ x \in ( a , b ) , $$

where the following conditions hold at the end points of the interval of orthogonality:

$$ \lim\limits _ {x \rightarrow a + 0 } \ \phi ( x) B ( x) = \ \lim\limits _ {x \rightarrow b - 0 } \ \phi ( x) B ( x) = 0 . $$

2) The polynomial $ y = P _ {n} ( x) $ of order $ n $ satisfies the differential equation

$$ B ( x) y ^ {\prime\prime} + [ A ( x) + B ^ { \prime } ( x) ] y ^ \prime - n [ p _ {1} + ( n + 1 ) q _ {2} ] y = 0 . $$

3) The Rodrigues formula holds:

$$ P _ {n} ( x) = \ \frac{c _ {n} }{\phi ( x) } \frac{d ^ {n} }{d x ^ {n} } [ \phi ( x) B ^ {n} ( x) ] , $$

where $ c _ {n} $ is a normalizing coefficient.

4) Derivatives of classical orthogonal polynomials are also classical orthogonal polynomials and are orthogonal on the same interval of orthogonality, generally speaking with a different weight.

5) For the generating function

$$ F ( x , w ) = \ \sum _ { n= 0} ^ \infty \frac{P _ {n} ( x) }{n ! c _ {n} } w ^ {n} ,\ \ x \in ( a , b ) , $$

the representation

$$ F ( x , w ) = \ \frac{1}{\phi ( x) } \frac{\phi ( \lambda ) }{1 - w B ^ { \prime } ( \lambda ) } ,\ \ x \in ( a , b ) , $$

holds, where $ \lambda = \lambda ( x , w ) $ is the root of the quadratic equation $ \zeta - x - w B ( \zeta ) = 0 $ that is nearest to $ x $ for small $ | w | $.

Only the three systems of orthogonal polynomials mentioned satisfy these properties; for systems obtained from these three by linear transformations of the independent variable these properties also hold.

In the generalized Rodrigues formula, the normalizing coefficient $ c _ {n} $ is usually chosen by three different methods with the aim of either obtaining orthonormal polynomials, orthogonal polynomials with unit leading coefficient or so-called standardized orthogonal polynomials, for which the main formulas have the simplest form and which prove to be most convenient in applications.

The classical orthogonal polynomials are the eigen functions of certain eigen value problems for equations of Sturm–Liouville type. In these problems, each system of orthogonal polynomials (Jacobi polynomials, Hermite polynomials and Laguerre polynomials) is the unique sequence of solutions of the corresponding system of equations (see [4]).

Particular cases of the classical orthogonal polynomials are defined by the following choices of weight function and interval of orthogonality:

1) The Jacobi polynomials $ \{ P _ {n} ( x ; \alpha , \beta ) \} $ are orthogonal on the interval $ [ - 1 , 1 ] $ with weight $ \phi ( x) = ( 1 - x ) ^ \alpha ( 1 + x ) ^ \beta $, where $ \alpha , \beta > - 1 $. In particular, the case $ \alpha = \beta $ gives the ultraspherical polynomials or Gegenbauer polynomials $ \{ P _ {n} ( x ; \alpha ) \} $. The Legendre polynomials $ \{ P _ {n} ( x) \} $ correspond to the values $ \alpha = \beta = 0 $ and are orthogonal on $ [ - 1 , 1 ] $ with weight $ \phi ( x) \equiv 1 $. If $ \alpha = \beta = - 1 / 2 $, that is, $ \phi ( x) = [ ( 1 - x ) ( 1 + x ) ] ^ {1/2} $, then one obtains the Chebyshev polynomials of the first kind, $ \{ T _ {n} ( x) \} $, while for $ \alpha = \beta = 1 / 2 $, the Chebyshev polynomials of the second kind, $ \{ U _ {n} ( x) \} $, are obtained.

2) The Hermite polynomials $ \{ H _ {n} ( x) \} $ are orthogonal on $ ( - \infty , \infty ) $ with weight $ \phi ( x) = e ^ {- x ^ {2} } $.

3) The Laguerre polynomials $ \{ L _ {n} ( x ; \alpha ) \} $ are orthogonal on $ ( 0 , \infty ) $ with weight $ \phi ( x) = x ^ \alpha e ^ {-x} $, where $ \alpha > - 1 $.

See also the references to Orthogonal polynomials.

References

[1]	Ya.L. Geronimus, "Theory of orthogonal polynomials. A survey of the achievements in Soviet mathematics" , Moscow-Leningrad (1950) (In Russian)
[2]	H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[3]	D. Jackson, "Fourier series and orthogonal polynomials" , Carus Math. Monogr. , 6 , Math. Assoc. Amer. (1971)
[4]	A.F. Nikiforov, V.B. Uvarov, "Fundamentals of the theory of special functions" , Moscow (1974) (In Russian)
[5]	P.K. Suetin, "Classical orthogonal polynomials" , Moscow (1978) (In Russian)

Comments

The classical orthogonal polynomials and the systems obtained from them by linear transformations of the independent variable can be characterized as the systems of orthogonal polynomials which satisfy any one of the following three properties (cf. [a4]):

1) the derivatives of the polynomials again form a system of orthogonal polynomials;

2) the polynomials are the eigen functions of a linear second-order differential operator;

3) a Rodrigues formula (see main text) holds, where $ B $ is some polynomial.

More general orthogonal polynomials of classical type occur if differentiations are replaced by finite differences or $ q $- differences, cf. [a1] and the chart of the classical hypergeometric orthogonal polynomials in [a3].

More common notations are $ P _ {n} ^ {( \alpha , \beta ) } ( x) $ for the Jacobi polynomials, $ P _ {n} ^ {( \alpha ) } ( x) $ or $ C _ {n} ^ {\alpha + 1 / 2 } ( x) $ for the Gegenbauer polynomials, and $ L _ {n} ^ \alpha ( x) $ for the Laguerre polynomials. Laguerre and Hermite polynomials can be obtained as limit cases of Jacobi polynomials. Jacobi and Laguerre polynomials can be written as terminating hypergeometric series of type $ {} _ {2} F _ {1} $, respectively $ {} _ {1} F _ {1} $.

The classical orthogonal polynomials have numerous applications in many branches of mathematics, in physics and in other sciences. These polynomials also have significant group-theoretic interpretations. The harmonic analysis of series expansions in terms of classical orthogonal polynomials is well-known and serves as a proto-type for harmonic analysis with more general orthogonal polynomials, cf. [a5].

References

[a1]	R.A. Askey, "Classical orthogonal polynomials" C. Brezinski (ed.) A. Draux (ed.) A.P. Magnus (ed.) P. Maroni (ed.) A. Ronveaux (ed.) , Polynômes Orthogonaux et Applications (Bar-le-Duc, 1984) , Lect. notes in math. , 1171 , Springer (1985) pp. 36–62
[a2]	R. Askey, "Orthogonal polynomials and special functions" , Reg. Conf. Ser. Appl. Math. , 21 , SIAM (1975)
[a3]	R. Askey, J. Wilson, "Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials" , Amer. Math. Soc. (1985)
[a4]	T.S. Chihara, "An introduction to orthogonal polynomials" , Gordon & Breach (1978 Zbl 0389.33008)
[a5]	G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)