Ultraspherical Polynomial

(or Gegenbauer polynomial). Ultraspherical polynomials are a special family of polynomials of successively increasing degree. For n = 0,1, 2,. . ., the ultraspherical polynomials of degree n are the coefficients of aⁿ in the power series expansion of the function

Ultraspherical polynomials are orthogonal on the interval [–1, + 1] with respect to the weight function (1 – x²)^λ–(½). Ultra spherical polynomials constitute a special case of Jacobi polynomials.