"게겐바워 다항식(ultraspherical polynomials)"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
||
| (같은 사용자의 중간 판 3개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
==개요== | ==개요== | ||
| − | * 직교다항식 | + | * 직교다항식 <math>C_n^{(\lambda )}(x)</math> |
| − | * [[자코비 다항식]] | + | * [[자코비 다항식]] <math>P_{n}^{(\alpha\,\beta)}(x)</math>의 특수한 경우 |
| − | + | :<math> | |
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) | 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) | ||
| − | + | </math> | |
==테이블== | ==테이블== | ||
| − | + | :<math> | |
\begin{array}{c|c} | \begin{array}{c|c} | ||
n & C_n{}^{(\lambda )}(x) \\ | n & C_n{}^{(\lambda )}(x) \\ | ||
| 19번째 줄: | 19번째 줄: | ||
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) \\ | 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} | \end{array} | ||
| − | + | </math> | |
| 27번째 줄: | 27번째 줄: | ||
==관련논문== | ==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. | * 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. | ||
[[분류:특수함수]] | [[분류:특수함수]] | ||
2020년 11월 13일 (금) 05:55 기준 최신판
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
\[ \begin{array}{c|c} n & C_n{}^{(\lambda )}(x) \\ \hline 0 & 1 \\ 1 & 2 \lambda x \\ 2 & \lambda \left(2 (\lambda +1) x^2-1\right) \\ 3 & \frac{2}{3} \lambda (\lambda +1) x \left(2 (\lambda +2) x^2-3\right) \\ 4 & \frac{1}{6} \lambda (\lambda +1) \left(4 (\lambda +2) (\lambda +3) x^4-12 (\lambda +2) x^2+3\right) \\ 5 & \frac{1}{15} \lambda (\lambda +1) (\lambda +2) x \left(4 (\lambda +3) (\lambda +4) x^4-20 (\lambda +3) x^2+15\right) \\ \end{array} \]
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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.