Chapter 10: Processes of Perception and Analysis

Section 11: Traditional Mathematics and Mathematical Formulas

Gegenbauer functions

Introduced by Leopold Gegenbauer in 1893 GegenbauerC[n, m, z] is a polynomial in z with integer coefficients for all integer n and m. It is a special case of Hypergeometric2F1 and JacobiP and satisfies a second-order ordinary differential equation in z. The GegenbauerC[n, d/2-1, z] form a set of orthogonal functions on a d-dimensional sphere. The GegenbauerC[n, 1/2, z] obtained for d=3 are LegendreP[n, z].

From Stephen Wolfram: A New Kind of Science [citation]