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| − | The basic hypergeometric series ${}_j\phi{}_{\ell}$ is defined by | + | The basic hypergeometric series ${}_r\phi{}_s$ is defined by |
| − | $${\small{}_j\phi_{\ell}(a_1,\ldots,a_j;b_1,\ldots,b_{\ell};q,z)={}_j \phi_{\ell} \left[ \begin{array}{llllll} | + | $${}_r \phi_s(a_1,a_2,\ldots,a_r; b_1,b_2,\ldots,b_s; z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k(a_2;q)_k \ldots (a_r;q)_k}{(b_1;q)_k (b_2;q)_k \ldots (b_s;q)_k} \dfrac{z^k}{(q;q)_k},$$ |
| − | a_1 & a_2 & \ldots & a_j \\
| + | where $(a_1;q)_k$ denotes the [[q-shifted factorial]]. |
| − | & & & & ; q,z \\
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| − | b_1 & b_2 & \ldots & b_{\ell}
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| − | \end{array}\right]=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k \ldots (a_j;q)_k}{(b_1;q)_k \ldots (b_{\ell};q)_k} \left((-1)^kq^{k \choose 2} \right)^{1+\ell-j}z^k=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k \ldots (a_j;q)_k}{(b_1;q)_k \ldots (b_{\ell};q)_k} \left(-q^{\frac{k-1}{2}} \right)^{k(1+\ell-j)}z^k.}$$
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| | =Properties= | | =Properties= |
| − | <div class="toccolours mw-collapsible mw-collapsed"> | + | [[Exponential e in terms of basic hypergeometric phi]]<br /> |
| − | <strong>Theorem:</strong> The following formula holds:
| + | [[1Phi0(a;;z) as infinite product]]<br /> |
| − | $$\displaystyle\lim_{q \rightarrow 1^-} {}_j \phi_{\ell} \left[ \begin{array}{l|l}
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| − | q^{a_1}, \ldots, q^{a_j} \\
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| − | q^{b_1}, \ldots, q^{b_{\ell}}
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| − | \end{array} \Bigg| q,z(1-q)^{1+\ell-j} \right]={}_j F_{\ell}\left(a_1,\ldots,a_j;b_1,\ldots,b_{\ell};(-1)^{1+\ell-j}z \right)$$
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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> | |
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| − | <div class="toccolours mw-collapsible mw-collapsed">
| + | =References= |
| − | <strong>Theorem:</strong> ($q$-Pfaff-Saalschütz) The following formula holds:
| + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=q-shifted factorial|next=1Phi0(a;;z) as infinite product}}: $4.8 (3)$ |
| − | $${}_3\phi_2(q^{-n},a,b;c,d;q,q) = \dfrac{\left(\frac{d}{a};q \right)_n \left( \frac{d}{b};q \right)_n}{\left(d;q \right)_n \left(\frac{d}{ab};q \right)_n},$$
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| − | with $cd=abq^{1-n}$.
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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>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> ($q$-Chu-Vandermonde sum) The following formula holds:
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| − | $${}_2\phi_1(q^{-n},a;c;q,q) = \dfrac{\left( \frac{c}{a};q \right)_n}{(c;q)_n} a^n.$$
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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>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> ($q$-Gauss theorem) The following formula holds:
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| − | $${}_2\phi_1 \left( a,b;c;q,\dfrac{c}{ab} \right) = \dfrac{\left(\frac{c}{a};q \right)_{\infty} \left( \frac{c}{b};q \right)_{\infty}}{\left(c;q\right)_{\infty} \left(\frac{c}{ab};q \right)_{\infty}}; \left| \dfrac{c}{ab} \right|<1.$$
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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>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> If $|z|<1$ or $a=q^{-n}$ then
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| − | $${}_1\phi_0(a;-;q,z) = \dfrac{(az;q)_{\infty}}{(z;q)_{\infty}}.$$
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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>
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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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| − | $${}_3\phi_2 (q^{-n},a,b;c,d;q,q) = \dfrac{b^n (\frac{d}{b};q)_n}{(d;q)_n} {}_3 \phi_2 \left(q^{-n},b,\frac{c}{a};c,q^{1-n}\frac{b}{d};q,\frac{aq}{d} \right).$$
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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>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> (Heine Transformation) The following formula holds for $|z|<1$ and $|b|<1$:
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| − | $${}_2\phi_1(a,b;c;q,z)=\dfrac{(b;q)_{\infty} (az;q)_{\infty}}{(c;q)_{\infty} (z;q)_{\infty}} {}_2\phi_1 \left( \frac{c}{b},z; az; q,b \right).$$
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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>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> The following formula holds for $|z|<1$ and $|abz|<|c|$:
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| − | $${}_2\phi_1(a,b;c;q,z) = \dfrac{(\frac{abz}{c};q)_{\infty})}{(z;q)_{\infty}} {}_2\phi_1 \left(\frac{c}{a},\frac{c}{b};c;q,\frac{abz}{c}\right).$$
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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>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> (Bailey-Daum sum) The following formula holds for $|q|<|b|$:
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| − | $${}_2\phi_1\left( a,b;\frac{aq}{b}; q, -\frac{q}{b} \right) = \dfrac{(-q;q)_{\infty} (aq;q^2)_{\infty} (\frac{aq^2}{b^2};q^2)_{\infty}}{(-\frac{q}{b};q)_{\infty} (\frac{aq}{b};q)_{\infty}}$$
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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>
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| | =See Also= | | =See Also= |
| | + | [[Hypergeometric pFq]]<br /> |
| | [[Basic hypergeometric series psi]] | | [[Basic hypergeometric series psi]] |
| | | | |
| − | {{:Exponential e in terms of basic hypergeometric phi}}
| + | [[Category:SpecialFunction]] |
The basic hypergeometric series ${}_r\phi{}_s$ is defined by
$${}_r \phi_s(a_1,a_2,\ldots,a_r; b_1,b_2,\ldots,b_s; z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k(a_2;q)_k \ldots (a_r;q)_k}{(b_1;q)_k (b_2;q)_k \ldots (b_s;q)_k} \dfrac{z^k}{(q;q)_k},$$
where $(a_1;q)_k$ denotes the q-shifted factorial.
