Gegenbauer Polynomial

A Gegenbauer Polynomial is an Orthogonal Polynomial that generalizes the Legendre and Chebyshev polynomials.

• AKA: Ultraspherical Polynomial.
• Context:
  • It is usually denoted $C_n^{\lambda}(x)$.
  • It is defined as the solutions to the Gegenbauer differential equation for integer $n$.
• Example(s):
  • $C_0^{\lambda}(x)=1$,
  • $C_1^{\lambda}(x)=2\lambda x$,
  • $C_2^{\lambda}(x)=-\lambda+2\lambda\left(1+\lambda\right)x^2$,
  • $C_3^{\lambda}(x)=-2\lambda\left(1+\lambda\right)x+\dfrac{4}{3}\lambda\left(1+\lambda\right)\left(2+\lambda\right)x^3$,
  • …
• Counter-Example(s):
  • Jacobi Polynomial,
  • Rogers Polynomial,
  • Romanovski Polynomial.
• See: Orthogonal Polynomials, Leopold Gegenbauer, Mathematics, Weight Function, Legendre Polynomials, Chebyshev Polynomials, Jacobi Polynomials.

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References

2021a

• (MathWorld, 2021) ⇒ Eric W. Weisstein (2021). "Gegenbauer Polynomial". In: MathWorld--A Wolfram Web Resource. Retrieved:2021-9-5.
  • QUOTE: The Gegenbauer polynomials $C_n^{\lambda}(x)$ are solutions to the Gegenbauer differential equation for integer $n$. They are generalizations of the associated Legendre polynomials to $\left(2\lambda+2\right)-D$ space, and are proportional to (or, depending on the normalization, equal to) the ultraspherical polynomials $P_n^{\lambda}(x)$.

2021b

• (Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Gegenbauer_polynomials Retrieved:2021-9-5.
  • In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − x²)^α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer.

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