The Gegenbauer (ultraspherical) polynomials
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orthpoly::gegenbauer(n,a,x) computes the
value of the n-th
degree Gegenbauer polynomial with parameter a at
the point x.
Evaluation for real floating-point values x from the interval [- 1.0, 1.0] is numerically stable. See Example 2.
Polynomial expressions are returned, if identifiers or indexed identifiers are specified:
orthpoly::gegenbauer(2, a, x)
orthpoly::gegenbauer(3, 2, x)
Using arithmetical expressions as input, the “values” of these polynomials are returned:
orthpoly::gegenbauer(2, 1, 3+2*I)
orthpoly::gegenbauer(3, 2, exp(x + 2))
“Arithmetical expressions” include numbers:
orthpoly::gegenbauer(2, a, sqrt(2)), orthpoly::gegenbauer(3, 0.4, 8 + I), orthpoly::gegenbauer(1000, -1/3, 0.3)
If the degree of the polynomial is a variable or expression,
orthpoly::gegenbauer returns itself symbolically:
orthpoly::gegenbauer(n, a, x)
If a floating-point value is desired, then a direct call such as
orthpoly::gegenbauer(200, 4, 0.3)
is appropriate and yields a correct result. One should not evaluate the symbolic polynomial at a floating-point value, because this may be numerically unstable:
G200 := orthpoly::gegenbauer(200, 4, x):
DIGITS := 10: evalp(G200, x = 0.3)
This result is caused by numerical round-off. Also with increased
a few leading digits are correct:
DIGITS := 20: evalp(G200, x = 0.3)
delete DIGITS, G200:
A nonnegative integer or an arithmetical expression representing a nonnegative integer: the degree of the polynomial.
An arithmetical expression.
An indeterminate or an arithmetical expression. An indeterminate
is either an identifier (of domain type
The value of the Gegenbauer polynomial at point
returned as an arithmetical expression. If
an arithmetical expression, then
The Gegenbauer polynomials are given by the recursion formula
with G(0, a, x) = 1, G(1, a, x) = 2 a x.
For fixed real these polynomials are orthogonal on the interval [- 1, 1] with respect to the weight function .
coincides with the Legendre polynomial P(n, x).
G(n, 1, x) coincides with the Chebyshev polynomial U(n, x) of the second kind.
The polynomials G(n, 0, x) are trivial.