게겐바워 다항식(ultraspherical polynomials)

개요

• 직교다항식 \(C_n^{(\lambda )}(x)\)
• 자코비 다항식 \(P_{n}^{(\alpha\,\beta)}(x)\)의 특수한 경우

\[ C_n^{(\lambda )}(x)=\frac{(2 \lambda)_{n}}{\left(\lambda +\frac{1}{2}\right)_n}P_n^{\left(\lambda -\frac{1}{2},\lambda -\frac{1}{2}\right)}(x) \]

테이블

\[ \begin{array}{c|c} n & C_n{}^{(\lambda )}(x) \\ \hline 0 & 1 \\ 1 & 2 \lambda x \\ 2 & \lambda \left(2 (\lambda +1) x^2-1\right) \\ 3 & \frac{2}{3} \lambda (\lambda +1) x \left(2 (\lambda +2) x^2-3\right) \\ 4 & \frac{1}{6} \lambda (\lambda +1) \left(4 (\lambda +2) (\lambda +3) x^4-12 (\lambda +2) x^2+3\right) \\ 5 & \frac{1}{15} \lambda (\lambda +1) (\lambda +2) x \left(4 (\lambda +3) (\lambda +4) x^4-20 (\lambda +3) x^2+15\right) \\ \end{array} \]

매스매티카 파일 및 계산리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxYTRkelY1aWpIT0k/edit

관련논문

• Berg, Christian, and Emilio Porcu. ‘From Schoenberg Coefficients to Schoenberg Functions’. arXiv:1505.05682 [math], 21 May 2015. http://arxiv.org/abs/1505.05682.
• Guella, Jean C., and Valdir A. Menegatto. ‘Strictly Positive Definite Kernels on a Product of Spheres’. arXiv:1505.03695 [math], 14 May 2015. http://arxiv.org/abs/1505.03695.
• Belton, Alexander, Dominique Guillot, Apoorva Khare, and Mihai Putinar. ‘Schoenberg’s Positivity Theorem in Fixed Dimension’. arXiv:1504.07674 [math], 28 April 2015. http://arxiv.org/abs/1504.07674.
• Guella, J. C., V. A. Menegatto, and Ana P. Peron. “An Extension of a Theorem of Schoenberg to Products of Spheres.” arXiv:1503.08174 [math], March 27, 2015. http://arxiv.org/abs/1503.08174.