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''Gegenbauer polynomials''
 
''Gegenbauer polynomials''
  
[[Orthogonal polynomials|Orthogonal polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095030/u0950301.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095030/u0950302.png" /> with the weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095030/u0950303.png" />; a particular case of the [[Jacobi polynomials|Jacobi polynomials]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095030/u0950304.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095030/u0950305.png" />); the [[Legendre polynomials|Legendre polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095030/u0950306.png" /> are a particular case of the ultraspherical polynomials: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095030/u0950307.png" />.
+
[[Orthogonal polynomials|Orthogonal polynomials]] $  P _ {n} ( x, \lambda ) $
 +
on the interval $  [ - 1 , 1 ] $
 +
with the weight function $  h ( x) = ( 1 - x  ^ {2} ) ^ {\lambda - 1 / 2 } $;  
 +
a particular case of the [[Jacobi polynomials|Jacobi polynomials]] for $  \alpha = \beta = \lambda - 1 / 2 $(
 +
$  \lambda > - 1 / 2 $);  
 +
the [[Legendre polynomials|Legendre polynomials]] $  P _ {n} ( x) $
 +
are a particular case of the ultraspherical polynomials: $  P _ {n} ( x) = P _ {n} ( x , 1 / 2 ) $.
  
 
For ultraspherical polynomials one has the standardization
 
For ultraspherical polynomials one has the standardization
  
<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/u/u095/u095030/u0950308.png" /></td> </tr></table>
+
$$
 +
P _ {n} ( x , \lambda )  \equiv \
 +
C _ {n} ^ {( \lambda ) } ( x) =
 +
$$
  
<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/u/u095/u095030/u0950309.png" /></td> </tr></table>
+
$$
 +
= \
  
<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/u/u095/u095030/u09503010.png" /></td> </tr></table>
+
\frac{( - 2 )  ^ {n} }{n!}
 +
 +
\frac{\Gamma ( n + \lambda ) \Gamma ( n
 +
+ 2 \lambda ) }{\Gamma ( \lambda ) \Gamma ( 2 n + 2 \lambda )
 +
}
 +
( 1 - x  ^ {2} ) ^ {- \lambda + 1 / 2 } \times
 +
$$
 +
 
 +
$$
 +
\times
 +
 
 +
\frac{d  ^ {n} }{d x  ^ {n} }
 +
[ ( 1 - x  ^ {2} ) ^ {n + \lambda - 1 / 2 } ]
 +
$$
  
 
and the representation
 
and the representation
  
<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/u/u095/u095030/u09503011.png" /></td> </tr></table>
+
$$
 +
C _ {n} ^ {( \lambda ) } ( x)  = \
 +
\sum _ { k= 0} ^ { [  n / 2 ] } ( - 1 )  ^ {k}
 +
 
 +
\frac{\Gamma ( n - k + \lambda ) }{\Gamma ( \lambda ) k ! ( n - 2 k ) ! }
 +
 
 +
( 2 x )  ^ {n-} 2k .
 +
$$
  
 
The ultraspherical polynomials are the coefficients of the power series expansion of the generating function
 
The ultraspherical polynomials are the coefficients of the power series expansion of the generating 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/u/u095/u095030/u09503012.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{1}{( 1 - 2 x w + w  ^ {2} )  ^  \lambda  }
 +
  = \
 +
\sum _ { n= 0} ^  \infty 
 +
C _ {n} ^ {( \lambda ) } ( x) w  ^ {n} .
 +
$$
  
The ultraspherical polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095030/u09503013.png" /> satisfies the differential equation
+
The ultraspherical polynomial $  C _ {n} ^ {( \lambda ) } ( 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/u/u095/u095030/u09503014.png" /></td> </tr></table>
+
$$
 +
( 1 - x  ^ {2} ) y  ^ {\prime\prime} -
 +
( 2 \lambda + 1 ) x y  ^  \prime
 +
+ n ( n + 2 \lambda ) y  = 0 .
 +
$$
  
 
More commonly used are the formulas
 
More commonly used are the formulas
  
<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/u/u095/u095030/u09503015.png" /></td> </tr></table>
+
$$
 +
( n + 1 ) C _ {n+1} ^ {( \lambda ) } ( x)  = \
 +
2 ( n + \lambda ) x C _ {n} ^ {( \lambda ) } ( x) -
 +
( n + 2 \lambda - 1 ) C _ {n-1} ^ {( \lambda ) } ( x) ,
 +
$$
  
<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/u/u095/u095030/u09503016.png" /></td> </tr></table>
+
$$
 +
C _ {n} ^ {( \lambda ) } ( - x )  = ( - 1 )  ^ {n} C _ {n} ^ {( \lambda ) } ( x) ,
 +
$$
  
<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/u/u095/u095030/u09503017.png" /></td> </tr></table>
+
$$
  
<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/u/u095/u095030/u09503018.png" /></td> </tr></table>
+
\frac{d}{dx}
 +
[ C _ {n} ^ {( \lambda ) } ( x) ]
 +
= 2 \lambda C _ {n-1} ^ {( \lambda + 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/u/u095/u095030/u09503019.png" /></td> </tr></table>
+
$$
 +
C _ {n} ^ {( \lambda ) } ( \pm  1 )  = ( \pm  1 )  ^ {n}
  
For references see [[Orthogonal polynomials|Orthogonal polynomials]].
+
\frac{2 \lambda ( 2 \lambda + 1 ) \dots ( 2 \lambda + n - 1 ) }{n!\ }
 +
=
 +
$$
  
 +
$$
 +
= \
 +
( \pm  1 )  ^ {n} \left ( \begin{array}{c}
 +
n + 2 \lambda - 1 \\
 +
n
 +
\end{array}
 +
\right ) .
 +
$$
  
 +
For references see [[Orthogonal polynomials|Orthogonal polynomials]].
  
 
====Comments====
 
====Comments====
 
See [[Spherical harmonics|Spherical harmonics]] for a group-theoretic interpretation. Ultraspherical polynomials are also connected with Jacobi polynomials by the quadratic transformations
 
See [[Spherical harmonics|Spherical harmonics]] for a group-theoretic interpretation. Ultraspherical polynomials are also connected with Jacobi polynomials by the quadratic transformations
  
<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/u/u095/u095030/u09503020.png" /></td> </tr></table>
+
$$
 +
C _ {2n} ^ {( \lambda ) } ( x)  = \
 +
\textrm{ const }  P _ {n} ^ {( \lambda - 1/2 , - 1/2) } ( 2x  ^ {2} - 1) ,
 +
$$
  
<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/u/u095/u095030/u09503021.png" /></td> </tr></table>
+
$$
 +
C _ {2n+ 1 }  ^ {( \lambda ) } ( x)  = \textrm{ const }  x P _ {n} ^ {( \lambda - 1/2, 1/2) } ( 2x  ^ {2} - 1) .
 +
$$
  
See [[#References|[a1]]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095030/u09503022.png" />-ultraspherical polynomials.
+
See [[#References|[a1]]] for $  q $-ultraspherical polynomials.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.A. Askey,  M.E.H. Ismail,  "A generalization of ultraspherical polynomials"  P. Erdös (ed.) , ''Studies in Pure Mathematics to the Memory of Paul Turán'' , Birkhäuser  (1983)  pp. 55–78</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.A. Askey,  M.E.H. Ismail,  "A generalization of ultraspherical polynomials"  P. Erdös (ed.) , ''Studies in Pure Mathematics to the Memory of Paul Turán'' , Birkhäuser  (1983)  pp. 55–78</TD></TR></table>

Latest revision as of 07:38, 26 February 2022


Gegenbauer polynomials

Orthogonal polynomials $ P _ {n} ( x, \lambda ) $ on the interval $ [ - 1 , 1 ] $ with the weight function $ h ( x) = ( 1 - x ^ {2} ) ^ {\lambda - 1 / 2 } $; a particular case of the Jacobi polynomials for $ \alpha = \beta = \lambda - 1 / 2 $( $ \lambda > - 1 / 2 $); the Legendre polynomials $ P _ {n} ( x) $ are a particular case of the ultraspherical polynomials: $ P _ {n} ( x) = P _ {n} ( x , 1 / 2 ) $.

For ultraspherical polynomials one has the standardization

$$ P _ {n} ( x , \lambda ) \equiv \ C _ {n} ^ {( \lambda ) } ( x) = $$

$$ = \ \frac{( - 2 ) ^ {n} }{n!} \frac{\Gamma ( n + \lambda ) \Gamma ( n + 2 \lambda ) }{\Gamma ( \lambda ) \Gamma ( 2 n + 2 \lambda ) } ( 1 - x ^ {2} ) ^ {- \lambda + 1 / 2 } \times $$

$$ \times \frac{d ^ {n} }{d x ^ {n} } [ ( 1 - x ^ {2} ) ^ {n + \lambda - 1 / 2 } ] $$

and the representation

$$ C _ {n} ^ {( \lambda ) } ( x) = \ \sum _ { k= 0} ^ { [ n / 2 ] } ( - 1 ) ^ {k} \frac{\Gamma ( n - k + \lambda ) }{\Gamma ( \lambda ) k ! ( n - 2 k ) ! } ( 2 x ) ^ {n-} 2k . $$

The ultraspherical polynomials are the coefficients of the power series expansion of the generating function

$$ \frac{1}{( 1 - 2 x w + w ^ {2} ) ^ \lambda } = \ \sum _ { n= 0} ^ \infty C _ {n} ^ {( \lambda ) } ( x) w ^ {n} . $$

The ultraspherical polynomial $ C _ {n} ^ {( \lambda ) } ( x) $ satisfies the differential equation

$$ ( 1 - x ^ {2} ) y ^ {\prime\prime} - ( 2 \lambda + 1 ) x y ^ \prime + n ( n + 2 \lambda ) y = 0 . $$

More commonly used are the formulas

$$ ( n + 1 ) C _ {n+1} ^ {( \lambda ) } ( x) = \ 2 ( n + \lambda ) x C _ {n} ^ {( \lambda ) } ( x) - ( n + 2 \lambda - 1 ) C _ {n-1} ^ {( \lambda ) } ( x) , $$

$$ C _ {n} ^ {( \lambda ) } ( - x ) = ( - 1 ) ^ {n} C _ {n} ^ {( \lambda ) } ( x) , $$

$$ \frac{d}{dx} [ C _ {n} ^ {( \lambda ) } ( x) ] = 2 \lambda C _ {n-1} ^ {( \lambda + 1) } ( x) , $$

$$ C _ {n} ^ {( \lambda ) } ( \pm 1 ) = ( \pm 1 ) ^ {n} \frac{2 \lambda ( 2 \lambda + 1 ) \dots ( 2 \lambda + n - 1 ) }{n!\ } = $$

$$ = \ ( \pm 1 ) ^ {n} \left ( \begin{array}{c} n + 2 \lambda - 1 \\ n \end{array} \right ) . $$

For references see Orthogonal polynomials.

Comments

See Spherical harmonics for a group-theoretic interpretation. Ultraspherical polynomials are also connected with Jacobi polynomials by the quadratic transformations

$$ C _ {2n} ^ {( \lambda ) } ( x) = \ \textrm{ const } P _ {n} ^ {( \lambda - 1/2 , - 1/2) } ( 2x ^ {2} - 1) , $$

$$ C _ {2n+ 1 } ^ {( \lambda ) } ( x) = \textrm{ const } x P _ {n} ^ {( \lambda - 1/2, 1/2) } ( 2x ^ {2} - 1) . $$

See [a1] for $ q $-ultraspherical polynomials.

References

[a1] R.A. Askey, M.E.H. Ismail, "A generalization of ultraspherical polynomials" P. Erdös (ed.) , Studies in Pure Mathematics to the Memory of Paul Turán , Birkhäuser (1983) pp. 55–78
How to Cite This Entry:
Ultraspherical polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ultraspherical_polynomials&oldid=14267
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article