Difference between revisions of "Basic hypergeometric phi"
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| − | The basic hypergeometric series ${} | + | 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},$$ |
| − | + | where $(a_1;q)_k$ denotes the [[q-shifted factorial]]. | |
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| − | |||
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=Properties= | =Properties= | ||
| − | < | + | [[Exponential e in terms of basic hypergeometric phi]]<br /> |
| − | < | + | [[1Phi0(a;;z) as infinite product]]<br /> |
| − | + | ||
| − | + | =References= | |
| − | + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=q-shifted factorial|next=1Phi0(a;;z) as infinite product}}: $4.8 (3)$ | |
| − | + | ||
| − | + | =See Also= | |
| − | < | + | [[Hypergeometric pFq]]<br /> |
| − | + | [[Basic hypergeometric series psi]] | |
| − | + | ||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 23:26, 3 March 2018
The basic hypergeometric series ${}_r\phi{}_s$ is defined by $${}_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
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $4.8 (3)$
