Gegenbauer polynomials wont produce Chebyshev polynomials using Symbolic Toolbox
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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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Risposte (1)
Sunand Agarwal
il 14 Ott 2020
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 Commenti
Sunand Agarwal
il 28 Ott 2020
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.
Hope this helps.
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