Difference between revisions of "Basic hypergeometric phi"

From specialfunctionswiki
Jump to: navigation, search
Line 4: Line 4:
 
     &    &        &    & ; q,z \\
 
     &    &        &    & ; q,z \\
 
b_1 & b_2 & \ldots & b_{\ell}
 
b_1 & b_2 & \ldots & b_{\ell}
\end{array}\right]=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k \ldots (a_j;q)_k}{(b_1;q)_k \ldots (b_{\ell};q)_k} \left((-1)^kq^{k \choose 2} \right)^{1+\ell-j}z^k.$$
+
\end{array}\right]=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k \ldots (a_j;q)_k}{(b_1;q)_k \ldots (b_{\ell};q)_k} \left((-1)^kq^{k \choose 2} \right)^{1+\ell-j}z^k=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k \ldots (a_j;q)_k}{(b_1;q)_k \ldots (b_{\ell};q)_k} \left(-q^{\frac{k-1}{2}} \right)^{k(1+\ell-j)}z^k.$$

Revision as of 15:48, 20 May 2015

The basic hypergeometric series ${}_j\phi{}_{\ell}$ is defined by $${}_j \phi_{\ell} \left[ \begin{array}{llllll} a_1 & a_2 & \ldots & a_j \\

   &     &        &     & ; q,z \\

b_1 & b_2 & \ldots & b_{\ell} \end{array}\right]=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k \ldots (a_j;q)_k}{(b_1;q)_k \ldots (b_{\ell};q)_k} \left((-1)^kq^{k \choose 2} \right)^{1+\ell-j}z^k=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k \ldots (a_j;q)_k}{(b_1;q)_k \ldots (b_{\ell};q)_k} \left(-q^{\frac{k-1}{2}} \right)^{k(1+\ell-j)}z^k.$$