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$$  
 
$$  
H _ {n+} 1 ( x)  =  2 x H _ {n} ( x) - 2 n H _ {n-} 1 ( x) ,
+
H _ {n+} 1 ( x)  =  2 x H _ {n} ( x) - 2 n H _ {n- 1} ( x) ,
 
$$
 
$$
  
 
$$  
 
$$  
H _ {n}  ^  \prime  ( x)  =  2 n H _ {n-} 1 ( x) ,
+
H _ {n}  ^  \prime  ( x)  =  2 n H _ {n- 1} ( x) ,
 
$$
 
$$
  
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H _ {n} ( x)  =  \sum _ {k=0} ^ { [ n/2] }
 
H _ {n} ( x)  =  \sum _ {k=0} ^ { [ n/2] }
 
\frac{(- 1)  ^ {k} n ! }{k ! ( n - 2 k ) ! }
 
\frac{(- 1)  ^ {k} n ! }{k ! ( n - 2 k ) ! }
  ( 2 x )  ^ {n-} 2k ,
+
  ( 2 x )  ^ {n-2k} ,
 
$$
 
$$
  
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\frac{(- 1)  ^ {n} }{2  ^ {n} }
 
\frac{(- 1)  ^ {n} }{2  ^ {n} }
 
  e ^ {x  ^ {2} }
 
  e ^ {x  ^ {2} }
( e ^ {- x  ^ {2} } )  ^ {(} n) .
+
( e ^ {- x  ^ {2} } )  ^ {(n) }.
 
$$
 
$$
  
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$$
 
$$
  
The definition of Hermite polynomials is encountered in P. Laplace [[#References|[1]]]. A detailed study of them was published by P.L. Chebyshev in 1859 (see [[#References|[2]]]). Later, these polynomials were studied by Ch. Hermite . V.A. Steklov [[#References|[4]]] proved that the set of them is dense in the space of square-summable functions with the weight  $  h ( x) =  \mathop{\rm exp} ( - x  ^ {2} ) $
+
The definition of Hermite polynomials is encountered in P. Laplace [[#References|[1]]]. A detailed study of them was published by [[Chebyshev, Pafnutii Lvovich|P.L. Chebyshev]] in 1859 (see [[#References|[2]]]). Later, these polynomials were studied by Ch. Hermite . V.A. Steklov [[#References|[4]]] proved that the set of them is dense in the space of square-summable functions with the weight  $  h ( x) =  \mathop{\rm exp} ( - x  ^ {2} ) $
 
on the whole real line.
 
on the whole real line.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Laplace,  ''Mém. Cl. Sci. Math. Phys. Inst. France'' , '''58'''  (1810)  pp. 279–347</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.L. Chebyshev,  , ''Collected works'' , '''2''' , Moscow-Leningrad  (1947)  pp. 335–341  (In Russian)</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  Ch. Hermite,  ''C.R. Acad. Sci. Paris'' , '''58'''  (1864)  pp. 93–100</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  Ch. Hermite,  ''C.R. Acad. Sci. Paris'' , '''58'''  (1864)  pp. 266–273</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Steklov,  ''Izv. Akad. Nauk'' , '''10'''  (1956)  pp. 403–416</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.K. Suetin,  "Classical orthogonal polynomials" , Moscow  (1979)  (In Russian)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Laplace,  ''Mém. Cl. Sci. Math. Phys. Inst. France'' , '''58'''  (1810)  pp. 279–347</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  P.L. Chebyshev,  , ''Collected works'' , '''2''' , Moscow-Leningrad  (1947)  pp. 335–341  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[3a]</TD> <TD valign="top">  Ch. Hermite,  ''C.R. Acad. Sci. Paris'' , '''58'''  (1864)  pp. 93–100</TD></TR>
 +
<TR><TD valign="top">[3b]</TD> <TD valign="top">  Ch. Hermite,  ''C.R. Acad. Sci. Paris'' , '''58'''  (1864)  pp. 266–273</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Steklov,  ''Izv. Akad. Nauk'' , '''10'''  (1956)  pp. 403–416</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top">  P.K. Suetin,  "Classical orthogonal polynomials" , Moscow  (1979)  (In Russian)</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====

Latest revision as of 07:13, 20 April 2024


Chebyshev–Hermite polynomials

Polynomials orthogonal on $ ( - \infty , \infty ) $ with the weight function $ h ( x) = e ^ {- x ^ {2} } $. The standardized Hermite polynomials are defined by the Rodrigues formula

$$ H _ {n} ( x) = ( - 1 ) ^ {n} e ^ {x ^ {2} } ( e ^ {- x ^ {2} } ) ^ {( n)} . $$

The most commonly used formulas are:

$$ H _ {n+} 1 ( x) = 2 x H _ {n} ( x) - 2 n H _ {n- 1} ( x) , $$

$$ H _ {n} ^ \prime ( x) = 2 n H _ {n- 1} ( x) , $$

$$ H _ {n} ( x) = \sum _ {k=0} ^ { [ n/2] } \frac{(- 1) ^ {k} n ! }{k ! ( n - 2 k ) ! } ( 2 x ) ^ {n-2k} , $$

$$ \mathop{\rm exp} ( 2 x w - w ^ {2} ) = \sum _ {n=0}^ \infty \frac{H _ {n} ( x) }{n!} w ^ {n} . $$

The first few Hermite polynomials are:

$$ H _ {0} ( x) = 1,\ H _ {1} ( x) = 2x,\ H _ {2} ( x) = 4 x ^ {2} - 2 , $$

$$ H _ {3} ( x) = 8 x ^ {3} - 12 x ,\ H _ {4} ( x) = 16 x ^ {4} - 48 x ^ {2} + 12 , $$

$$ H _ {5} ( x) = 32 x ^ {5} + 160 x ^ {3} + 120 x ,\dots . $$

The polynomial $ H _ {n} ( x) $ satisfies the differential equation

$$ y ^ {\prime\prime} - 2 x y ^ \prime + 2 n y = 0 . $$

The orthonormal Hermite polynomials are defined by

$$ \widehat{H} _ {n} ( x) = \frac{H _ {n} ( x) }{\sqrt {n ! 2 ^ {n} \sqrt \pi } } . $$

The Hermite polynomials with leading coefficient one have the form

$$ \widetilde{H} _ {n} ( x) = \frac{1}{2 ^ {n} } H _ {n} ( x) = \ \frac{(- 1) ^ {n} }{2 ^ {n} } e ^ {x ^ {2} } ( e ^ {- x ^ {2} } ) ^ {(n) }. $$

Fourier series in Hermite polynomials in the interior of $ ( - \infty , \infty ) $ behave analogous to trigonometric Fourier series.

In mathematical statistics and probability theory one uses the Hermite polynomials corresponding to the weight function

$$ h ( x) = \mathop{\rm exp} ( - x ^ {2} / 2 ) . $$

The definition of Hermite polynomials is encountered in P. Laplace [1]. A detailed study of them was published by P.L. Chebyshev in 1859 (see [2]). Later, these polynomials were studied by Ch. Hermite . V.A. Steklov [4] proved that the set of them is dense in the space of square-summable functions with the weight $ h ( x) = \mathop{\rm exp} ( - x ^ {2} ) $ on the whole real line.

References

[1] P.S. Laplace, Mém. Cl. Sci. Math. Phys. Inst. France , 58 (1810) pp. 279–347
[2] P.L. Chebyshev, , Collected works , 2 , Moscow-Leningrad (1947) pp. 335–341 (In Russian)
[3a] Ch. Hermite, C.R. Acad. Sci. Paris , 58 (1864) pp. 93–100
[3b] Ch. Hermite, C.R. Acad. Sci. Paris , 58 (1864) pp. 266–273
[4] V.A. Steklov, Izv. Akad. Nauk , 10 (1956) pp. 403–416
[5] P.K. Suetin, "Classical orthogonal polynomials" , Moscow (1979) (In Russian)

Comments

The result by Steklov mentioned in the last sentence of the main article goes back at least to H. Weyl (1908), cf. the references in [a3], Sect. 5.7.

One possible way to prove the Plancherel formula for the Fourier transform is by use of Hermite polynomials, cf. [a4]. Hermite polynomials occur in solutions of the heat and Schrödinger equations and in the so-called heat polynomials, cf. [a1]. A canonical orthonormal basis of the representation space for the Schrödinger representation of the Heisenberg group is given in terms of Hermite polynomials, cf. [a2].

References

[a1] W. Miller jr., "Symmetry and separation of variables" , Addison-Wesley (1977)
[a2] W. Schempp, "Harmonic analysis on the Heisenberg nilpotent Lie group" , Longman (1986)
[a3] G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
[a4] E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948)
How to Cite This Entry:
Hermite polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_polynomials&oldid=54852
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article