Gegenbauer polynomials wont produce Chebyshev polynomials using Symbolic Toolbox

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jf
jf 2020-10-11
评论： Sunand Agarwal 2020-10-28
Consider code
syms x
n = 4;
a = -0.5;
gegenbauerC(n,a,x)
It produces following output
- (5*x^4)/8 + (3*x^2)/4 - 1/8
which is not correct. Expected result according to theory of orthogonal polynomials is
8*x^4 - 8*x^2 + 1
i.e. Chebyshev polynomial
chebyshevT(n,x)
What I am missing?

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回答（1 个）

Sunand Agarwal
Sunand Agarwal 2020-10-14
Please refer to this article to understand the relationships between Gegenbauer and Chebyshev polynomials.
You will find that the relation between them is as follows:
T(n,x) = (n/2) * G(n,0,x)
Hope this helps.
  2 个评论
jf
jf 2020-10-16
syms x n
n = 4;
n/2*gegenbauerC(n,0,x) % returns 0
chebyshevT(n,x) % returns correct Chebyshev polynomial
Sunand Agarwal
Sunand Agarwal 2020-10-28
I understand that the relationship between the two polynomials in the previous article is incorrect and we apologize for the same. This is a documentation bug and we're currently working on it.
Meanwhile, as a workaround, the correct mathematical expression for the relation between GegenbauerC and ChebyshevT polynomials can be found in Eq 38 in http://files.ele-math.com/articles/jca-03-02.pdf, which says
chebyshevT(n,x) = (1/epsilon) * lim a->0 {(n+a)/a}*gegenbauerC(n,a,x)
where epsilon = 1 for n = 0, and epsilon = 2 otherwise.
You may refer to this rule for solving the limit and hence the equation.
Hope this helps.

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