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''Gegenbauer polynomials''

[[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

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

$$ 
= \ 

\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|Orthogonal polynomials]].

====Comments====
See [[Spherical harmonics|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 [[#References|[a1]]] for  $  q $-
ultraspherical polynomials.

====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>