Namespaces
Variants
Actions

Ultraspherical polynomials

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.


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=52128
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article