Difference between revisions of "Basic hypergeometric phi"

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The basic hypergeometric series ${}_j\phi{}_{\ell}$ is defined by
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The basic hypergeometric series ${}_r\phi{}_s$ is defined by
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$${}_j\phi_{\ell}(a_1,\ldots,a_j;b_1,\ldots,b_{\ell};q,z)={}_j \phi_{\ell} \left[ \begin{array}{llllll}
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$${}_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},$$
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a_1 & a_2 & \ldots & a_j \\
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where $(a_1;q)_k$ denotes the [[q-shifted factorial]].
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    &    &        &    & ; 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.$$
 
  
 
=Properties=
 
=Properties=
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<div class="toccolours mw-collapsible mw-collapsed">
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[[Exponential e in terms of basic hypergeometric phi]]<br />
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<strong>Theorem:</strong> The following formula holds:
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[[1Phi0(a;;z) as infinite product]]<br />
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$$\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">
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=References=
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<strong>Theorem:</strong> ($q$-Pfaff-Saalschütz) The following formula holds:
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* {{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}}: $4.8 (3)$
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$${}_3\phi_2(q^{-n},a,b;c,d;q,q) = \dfrac{\left(\frac{d}{a};q \right)_n \left( \frac{d}{b};q \right)_n}{\left(d;q \right)_n \left(\frac{d}{ab};q \right)_n},$$
 
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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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=See Also=
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<strong>Theorem:</strong> ($q$-Chu-Vandermonde sum) The following formula holds:
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[[Hypergeometric pFq]]<br />
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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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[[Basic hypergeometric series psi]]
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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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[[Category:SpecialFunction]]
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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>
 

Latest revision as of 23:26, 3 March 2018

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.

Properties

Exponential e in terms of basic hypergeometric phi
1Phi0(a;;z) as infinite product

References

See Also

Hypergeometric pFq
Basic hypergeometric series psi

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