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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]=