로저스-세괴 다항식 (Rogers-Szegő polynomials)

개요

• 직교다항식의 하나
• \(0\leq k\leq n\)인 정수 \(k,n\)에 대하여, Q-이항계수 (가우스 다항식)

\[ \begin{bmatrix} n\\ k\end{bmatrix}_{q}:=\frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}} \]

• 로저스-세괴 다항식 \(H_m(z;q),\, m\in \mathbb{Z}_{\geq 0}\)는 다음과 같이 정의된다

\begin{equation}\label{RS} H_m(z;q):=\sum_{k=0}^m z^k \begin{bmatrix} m\\ k\end{bmatrix}_{q} \end{equation}

성질

• 다음과 같은 3항 점화식을 만족

\[ H_{m+1}(z;q)=(1+z)H_m(z;q)-(1-q^m) z H_{m-1}(z;q) \] 이 때, \(H_{-1}=0\), \(H_0=1\)

• 직교성

\[ \frac{1}{2\pi}\int_{0}^{2\pi}H_{m}(-q^{-1/2}e^{i\theta};q)H_{n}(-q^{-1/2}e^{-i\theta};q)\theta_{3}(\theta,q)\,d\theta=\frac{(q;q)_m}{q^m}\delta_{m,n} \] 여기서 \(\theta_3(\theta;q)=\sum_{m=-\infty}^{\infty}q^{m^2/2}e^{i m \theta}\)는 자코비 세타함수

테이블

\begin{array}{c|c} n & H_n\text{(x)} \\ \hline 0 & 1 \\ 1 & z+1 \\ 2 & q z+z^2+z+1 \\ 3 & (z+1) \left(q^2 z+q z+z^2+1\right) \\ 4 & q^4 z^2+q^3 z^3+q^3 z^2+q^3 z+q^2 z^3+2 q^2 z^2+q^2 z+q z^3+q z^2+q z+z^4+z^3+z^2+z+1 \\ \end{array}

메모

\[ w(z;q)=|(zq^{1/2};q)_{\infty}|^2 \qquad (0<q<1). \]

매스매티카 파일 및 계산 리소스

• https://drive.google.com/file/d/0B8XXo8Tve1cxWUl5TjZVLW82V1k/view

사전 형태의 자료

• https://en.wikipedia.org/wiki/Rogers–Szegő_polynomials

관련논문

• Johann Cigler, Elementary observations on Rogers-Szegö polynomials, arXiv:1602.07850[math.CA], February 25 2016, http://arxiv.org/abs/1602.07850v2
• Cigler, Johann. “Elementary Observations on Rogers-Szeg"o Polynomials.” arXiv:1602.07850 [math], February 25, 2016. http://arxiv.org/abs/1602.07850.
• Jagannathan, R., and R. Sridhar. “(p,q)-Rogers-Szego Polynomial and the (p,q)-Oscillator.” arXiv:1005.4309 [math-Ph], May 24, 2010. http://arxiv.org/abs/1005.4309.
• Warnaar, S. Ole. “Rogers-Szego Polynomials and Hall-Littlewood Symmetric Functions.” arXiv:0708.3110 [math], August 22, 2007. http://arxiv.org/abs/0708.3110.
• Galetti, D., S. S. Mizrahi, and M. Ruzzi. “The Wigner Function Associated with the Rogers-Szegö Polynomials.” Journal of Physics A: Mathematical and General 37, no. 50 (December 17, 2004): L643. doi:10.1088/0305-4470/37/50/L01.
• Atakishiyev, N. M., and Sh M. Nagiyev. “On the Rogers-Szego Polynomials.” Journal of Physics A: Mathematical and General 27, no. 17 (September 7, 1994): L611. doi:10.1088/0305-4470/27/17/003.

메타데이터

위키데이터

• ID : Q414751

Spacy 패턴 목록

• [{'LEMMA': 'roger'}]