Gegenbauer polynomials wont produce Chebyshev polynomials using Symbolic Toolbox

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?

Answer 1

[Sunand Agarwal]

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.

[jf]

syms x n

n = 4;

n/2*gegenbauerC(n,0,x) % returns 0

chebyshevT(n,x) % returns correct Chebyshev polynomial

[Sunand Agarwal]

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.