This is documentation for Mathematica 3, which was
based on an earlier version of the Wolfram Language.
 GegenbauerC GegenbauerC[ n , m , x ] gives the Gegenbauer polynomial . GegenbauerC[ n , x ] gives the renormalized form . Mathematical function (see Section A.3.10). Explicit polynomials are given for integer n and for any m. satisfies the differential equation . The Gegenbauer polynomials are orthogonal on the interval with weight function , corresponding to integration over a unit hypersphere. GegenbauerC[ n , 0, x ] is always zero. GegenbauerC[ n , m , z ] has a branch cut discontinuity in the complex z plane running from to . See the Mathematica book: Section 3.2.9. See also: LegendreP, ChebyshevT, ChebyshevU. Further Examples Here are the first ten GegenbauerC polynomials. In[1]:= Out[1]//TableForm= The Gegenbauer polynomials are pairwise orthogonal with respect to the appropriate weight function. In[2]:= Out[2]= This is the derivative. In[3]:= Out[3]= This is the indefinite integral. In[4]:= Out[4]= This verifies the defining differential equation for n = 2 and m = 3. In[5]:= Out[5]= This is a series expansion around . In[6]:= Out[6]=