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''Chebyshev–Hermite polynomials''
 
''Chebyshev–Hermite polynomials''
  
Polynomials orthogonal on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h0470101.png" /> with the weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h0470102.png" />. The standardized Hermite polynomials are defined by the [[Rodrigues formula|Rodrigues formula]]
+
Polynomials orthogonal on $  ( - \infty , \infty ) $
 +
with the weight function $  h ( x) = e ^ {- x  ^ {2} } $.  
 +
The standardized Hermite polynomials are defined by the [[Rodrigues formula|Rodrigues formula]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h0470103.png" /></td> </tr></table>
+
$$
 +
H _ {n} ( x)  = ( - 1 )  ^ {n} e ^ {x  ^ {2} }
 +
( e ^ {- x  ^ {2} } )  ^ {(} n) .
 +
$$
  
 
The most commonly used formulas are:
 
The most commonly used formulas are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h0470104.png" /></td> </tr></table>
+
$$
 +
H _ {n+} 1 ( x)  = 2 x H _ {n} ( x) - 2 n H _ {n-} 1 ( x) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h0470105.png" /></td> </tr></table>
+
$$
 +
H _ {n}  ^  \prime  ( x)  = 2 n H _ {n-} 1 ( x) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h0470106.png" /></td> </tr></table>
+
$$
 +
H _ {n} ( x)  = \sum _ { k= } 0 ^ { [ }  n/2]
 +
\frac{(- 1)  ^ {k} n ! }{k ! ( n - 2 k ) ! }
 +
( 2 x )  ^ {n-} 2k ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h0470107.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm exp} ( 2 x w - w  ^ {2} )  = \sum _ { n= } 0 ^  \infty 
 +
\frac{H _ {n} ( x) }{n!}
 +
w  ^ {n} .
 +
$$
  
 
The first few Hermite polynomials are:
 
The first few Hermite polynomials are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h0470108.png" /></td> </tr></table>
+
$$
 +
H _ {0} ( x)  = 1,\  H _ {1} ( x)  = 2x,\  H _ {2} ( x)  = 4 x  ^ {2} - 2 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h0470109.png" /></td> </tr></table>
+
$$
 +
H _ {3} ( x)  = 8 x  ^ {3} - 12 x ,\  H _ {4} ( x)  = 16 x  ^ {4} - 48 x  ^ {2} + 12 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h04701010.png" /></td> </tr></table>
+
$$
 +
H _ {5} ( x)  = 32 x  ^ {5} + 160 x  ^ {3} + 120 x ,\dots .
 +
$$
  
The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h04701011.png" /> satisfies the differential equation
+
The polynomial $  H _ {n} ( x) $
 +
satisfies the differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h04701012.png" /></td> </tr></table>
+
$$
 +
y  ^ {\prime\prime} - 2 x y  ^  \prime  + 2 n y  = 0 .
 +
$$
  
 
The orthonormal Hermite polynomials are defined by
 
The orthonormal Hermite polynomials are defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h04701013.png" /></td> </tr></table>
+
$$
 +
\widehat{H}  _ {n} ( x)  =
 +
\frac{H _ {n} ( x) }{\sqrt {n ! 2  ^ {n} \sqrt \pi } }
 +
.
 +
$$
  
 
The Hermite polynomials with leading coefficient one have the form
 
The Hermite polynomials with leading coefficient one have the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h04701014.png" /></td> </tr></table>
+
$$
 +
\widetilde{H}  _ {n} ( x)  =
 +
\frac{1}{2  ^ {n} }
 +
H _ {n} ( x)  = \
  
Fourier series in Hermite polynomials in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h04701015.png" /> behave analogous to trigonometric Fourier series.
+
\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
 
In mathematical statistics and probability theory one uses the Hermite polynomials corresponding to the weight function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h04701016.png" /></td> </tr></table>
+
$$
 +
h ( x)  =   \mathop{\rm exp} ( - x  ^ {2} / 2 ) .
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047010/h04701017.png" /> on the whole real line.
+
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} ) $
 +
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 22:10, 5 June 2020


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=47215
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article