Gegenbauer polynomial

Gegenbauer polynomials with α = 1

Gegenbauer polynomials with α = 2

The Gegenbauer polynomials , also called ultra-spherical polynomials , are a set of orthogonal polynomials on the interval with the weighting function , with . They are named after the mathematician Leopold Gegenbauer and form the solution to the Gegenbauer differential equation . The polynomials have the form ${\displaystyle [-1,1]}$ ${\displaystyle (1-x^{2})^{\alpha -1/2}}$ ${\displaystyle \alpha >-1/2}$

${\displaystyle C_{n}^{(\alpha )}(z)={\frac {1}{\Gamma (\alpha )}}\sum _{m=0}^{\lfloor n/2\rfloor }(-1)^{m}{\frac {\Gamma (\alpha +n-m)}{m!(n-2m)!}}(2z)^{n-2m},}$

for , otherwise ${\displaystyle \alpha \neq 0}$

${\displaystyle C_{n}^{(0)}(z)=\sum _{m=0}^{\lfloor n/2\rfloor }(-1)^{m}{\frac {(n-m-1)!}{m!(n-2m)!}}(2z)^{n-2m},}$

They can also be represented by a hypergeometric function : ${\displaystyle {}_{2}F_{1}}$

${\displaystyle C_{n}^{(\alpha )}(z)={\frac {(2\alpha +n-1)!}{(2\alpha -1)!\,n!}}\,_{2}F_{1}\left(-n,2\alpha +n;\alpha +{\frac {1}{2}};{\frac {1-z}{2}}\right)}$

The value for is ${\displaystyle z=1}$

${\displaystyle C_{n}^{(\alpha )}(1)={n+2\alpha -1 \choose n}.}$

The first polynomials have the form:

${\displaystyle C_{0}^{(\alpha )}(z)=1}$

${\displaystyle C_{1}^{(\alpha )}(z)=2\alpha z}$

${\displaystyle C_{2}^{(\alpha )}(z)=-\alpha +2\alpha (1+\alpha )z^{2}}$

${\displaystyle C_{3}^{(\alpha )}(z)=-2\alpha (1+\alpha )z+4/3\alpha (1+\alpha )(2+\alpha )z^{3}}$

credentials

• Eric W. Weisstein : Gegenbauer Polynomial . In: MathWorld (English).
• Milton Abramowitz , Irene Stegun : Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . New York: Dover. ISBN 0-486-61272-4 , p. 774 .