Difference between revisions of "Basic hypergeometric phi"
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(Created page with "The basic hypergeometric series $\phi$ is defined by $${}_j \phi_k \left[ \begin{array}{llllll} a_1 & a_2 & \ldots & a_j \\ & & & & ; q,z \\ b_1 & b_2 & \ld...") |
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| − | The basic hypergeometric series $\phi$ is defined by | + | The basic hypergeometric series ${}_j\phi{}_{\ell}$ is defined by |
| − | $${}_j \ | + | $${}_j \phi_{\ell} \left[ \begin{array}{llllll} |
a_1 & a_2 & \ldots & a_j \\ | a_1 & a_2 & \ldots & a_j \\ | ||
& & & & ; q,z \\ | & & & & ; q,z \\ | ||
b_1 & b_2 & \ldots & b_{\ell} | b_1 & b_2 & \ldots & b_{\ell} | ||
\end{array}\right]=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k \ldots (a_j;q)_k}{(b_1;q)_k \ldots (b_{\ell};q)_k} \left((-1)^kq^{k \choose 2} \right)^{1+\ell-j}z^n.$$ | \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^n.$$ | ||
Revision as of 07:26, 6 May 2015
The basic hypergeometric series ${}_j\phi{}_{\ell}$ is defined by $${}_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^n.$$
