Gegenbauer polynomials

The Gegenbauer polynomials, also called ultra- spherical polynomials are a set of orthogonal polynomials on the interval with the weight function (1- x2) α -1/ 2, with α > -1 / 2 They are named after the mathematician Leopold Gegenbauer and form the solution of the Gegenbauer differential equation. The polynomials have the form

For α ≠ 0, otherwise

They can also be represented by a hypergeometric function 2F1:

The value for z = 1 is

The first polynomials have the form:

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
  • Analysis
Wolfram Research Abramowitz and Stegun
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