Difference between revisions of "Basic hypergeometric series psi"

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The bilateral basic hypergeometric series $\psi$ is defined by
 
The bilateral basic hypergeometric series $\psi$ is defined by
 
$${}_j\psi_{\ell}(a_1,\ldots,a_j;b_1,\ldots,b_k;q,z)=\displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(a_1;q)_k\ldots(a_j;q)_k}{(b_1;q)_k\ldots(b_{\ell};q)_k}\left( (-1)^k q^{ {k \choose 2} } \right)^{\ell-j}z^k.$$
 
$${}_j\psi_{\ell}(a_1,\ldots,a_j;b_1,\ldots,b_k;q,z)=\displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(a_1;q)_k\ldots(a_j;q)_k}{(b_1;q)_k\ldots(b_{\ell};q)_k}\left( (-1)^k q^{ {k \choose 2} } \right)^{\ell-j}z^k.$$
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=Properties=
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<div class="toccolours mw-collapsible mw-collapsed">
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<strong>Theorem:</strong> The following formula holds:
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$${}_1\psi_1(a,b;q,z) = \dfrac{\left(\frac{b}{a};q \right)_{\infty} (q,q)_{\infty} \left( \frac{q}{az};q \right)_{\infty} (az;q)_{\infty} }{(b;q)_{\infty} \left( \frac{b}{az};q \right)_{\infty} \left( \frac{q}{a};q \right)_{\infty} (z;q)_{\infty}}.$$
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<div class="mw-collapsible-content">
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<strong>Proof:</strong> █
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</div>
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</div>

Revision as of 16:35, 20 May 2015

The bilateral basic hypergeometric series $\psi$ is defined by $${}_j\psi_{\ell}(a_1,\ldots,a_j;b_1,\ldots,b_k;q,z)=\displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(a_1;q)_k\ldots(a_j;q)_k}{(b_1;q)_k\ldots(b_{\ell};q)_k}\left( (-1)^k q^{ {k \choose 2} } \right)^{\ell-j}z^k.$$

Properties

Theorem: The following formula holds: $${}_1\psi_1(a,b;q,z) = \dfrac{\left(\frac{b}{a};q \right)_{\infty} (q,q)_{\infty} \left( \frac{q}{az};q \right)_{\infty} (az;q)_{\infty} }{(b;q)_{\infty} \left( \frac{b}{az};q \right)_{\infty} \left( \frac{q}{a};q \right)_{\infty} (z;q)_{\infty}}.$$

Proof: