나스랄라-라만 적분

개요

• 애스키-윌슨 적분의 확장
• 나스랄라-라만 적분 (Nassrallah-Rahman trigonometric beta integral)

\begin{equation}\label{NR} \frac{(q,q)_\infty}{2} \int_{\mathbb{T}}\frac{(z \prod_{i=1}^5 t_i,q)_\infty (z^{-1} \prod_{i=1}^5 t_i,q)_\infty (z^2,q)_\infty (z^{-2},q)_\infty}{\prod_{i=1}^5 (t_i z)_\infty (t_i z^{-1})_\infty} \frac{dz}{2\pi i z} \ = \ \frac{\prod_{j=1}^5 (\frac{t_1 t_2 t_3 t_4 t_5}{t_j},q)_\infty}{\prod_{1 \leq i < j \leq 5} (t_i t_j,q)_\infty} \end{equation}

확장

정리 (Spiridonov).

Let $t_1, \dots ,t_6,p,q \in {\mathbb{C}}$ with $|t_1|, \dots , |t_6|,|p|,|q| <1$. Then \begin{equation} \label{betaint} \frac{(p;p)_\infty (q;q)_\infty}{2} \int_{\mathbb{T}} \frac{\prod_{i=1}^6 \Gamma(t_i z ;p,q)\Gamma(t_i z^{-1} ;p,q)}{\Gamma(z^{2};p,q) \Gamma(z^{-2};p,q)} \frac{dz}{2 \pi i z} = \prod_{1 \leq i < j \leq 6} \Gamma(t_i t_j;p,q), \end{equation} where the unit circle $\mathbb{T}$ is taken in the positive orientation and we imposed the balancing condition $\prod_{i=1}^6 t_i=pq$.

• $p \rightarrow 0$일 때, \ref{NR}을 얻는다

메모

• observed by Rahman in \cite{Rahman2} as a special case of the integral found in \cite{Nasrallah}

관련논문

• Nassrallah, B., and M. Rahman. “Projection Formulas, a Reproducing Kernel and a Generating Function for Q-Wilson Polynomials.” SIAM Journal on Mathematical Analysis 16, no. 1 (January 1, 1985): 186–97. doi:10.1137/0516014.
• Askey, Richard, and James Arthur Wilson. Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Vol. 319. American Mathematical Soc., 1985.