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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]= 
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