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
$${}_j \phi_{\ell} \left[ \begin{array}{llllll}
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$${}_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},$$
a_1 & a_2 & \ldots & a_j \\
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where $(a_1;q)_k$ denotes the [[q-shifted factorial]].
    &    &        &    & ; q,z \\
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b_1 & b_2 & \ldots & b_{\ell}
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=Properties=
\end{array}\right]=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k \ldots (a_j;q)_k}{(b_1;q)_k \ldots (b_{\ell};q)_k} \left((-1)^kq^{k \choose 2} \right)^{1+\ell-j}z^k.$$
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[[Exponential e in terms of basic hypergeometric phi]]<br />
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[[1Phi0(a;;z) as infinite product]]<br />
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=References=
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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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=See Also=
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[[Hypergeometric pFq]]<br />
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[[Basic hypergeometric series psi]]
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[[Category:SpecialFunction]]

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