Gegenbauer C
The Gegenbauer polynomial of degree $n$ and order $\lambda$ are orthogonal polynomials defined to be the coefficient of $t^n$ in the expansion of $\dfrac{1}{(1-2xt+t^2)^{\lambda}}$ in the sense that $$\dfrac{1}{(1-2xt+t^2)^{\lambda}} = \sum_{k=0}^{\infty} C_k^{\lambda}(x)t^k.$$
Contents
Properties
Theorem: The Gegenbauer polynomials satisfy the differential equation $$(1-x^2)\dfrac{d^2y}{dx^2} -(2\lambda+1) x \dfrac{dy}{dx} + n(n+2\lambda)y=0.$$
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Theorem: The following formula holds: $$C_n^{\lambda}(x) = \displaystyle\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} (-1)^k \dfrac{\Gamma(n-k+\lambda)}{\Gamma(\lambda)k!(n-2k)!} (2x)^{n-2k}.$$
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Theorem (Orthogonality): The following formula holds: $$\displaystyle\int_{-1}^1 (1-x^2)^{\lambda-\frac{1}{2}} C_n^{\lambda}(x)C_m^{\lambda}(x) dx = 2^{1-2\lambda} \pi \dfrac{\Gamma(n+2\lambda)}{(n+\lambda)(\Gamma(\lambda))^2\Gamma(n+1)}\delta_{mn},$$ where $\delta_{mn}=0$ if $m\neq 0$ and $\delta_{mn}=1$ when $m=n$.
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Theorem: The following formula holds: $$(n+2)C_{n+2}^{\lambda}(x)=2(\lambda+n+1)xC_{n+1}^{\lambda}(x)-(2\lambda+n)C_n^{\lambda}(x).$$
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Theorem: The following formula holds: $$nC_n^{\lambda}(x) = 2\lambda(xC_{n-1}^{\lambda+1}(x) - C_{n-2}^{\lambda+1}(x)).$$
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Theorem: The following formula holds: $$(n+2\lambda)C_n^{\lambda}(x) = 2\lambda(C_n^{\lambda+1}(x)-xC_{n-1}^{\lambda+1}(x))$$.
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Theorem: The following formula holds: $$nC_n^{\lambda}(x) = (n-1+2\lambda)xC_{n-1}^{\lambda}(x) - 2\lambda(1-x^2)C_{n-2}^{\lambda-1}(x).$$
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Theorem: The following formula holds: $$C_n^{\lambda '}(x) = 2\lambda C_{n+1}^{\lambda+1}(x).$$
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Theorem
The following formula holds: $$C_n^{\lambda}(x)=\dfrac{\Gamma(\lambda+\frac{1}{2})\Gamma(n+2\lambda)}{\Gamma(2\lambda)\Gamma(n+\lambda+\frac{1}{2})}P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x),$$ where $C_n$ denotes a Gegenbauer C polynomial and $P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}$ denotes a Jacobi P polynomial.
Proof
References
Theorem
The following formula holds for $n \in \{1,2,3,\ldots\}$: $$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda},$$ where $T_n$ denotes a Chebyshev polynomial of the first kind and $C_n^{\lambda}$ denotes a Gegenbauer C polynomial.
Proof
References
Theorem
The following formula holds for $n \in \{1,2,3,\ldots\}$: $$U_n(x)=\sqrt{1-x^2}C_{n-1}^1(x),$$ where $U_n$ denotes a Chebyshev polynomial of the second kind and $C_{n-1}^1$ denotes a Gegenbauer C polynomial.
