Difference between revisions of "Hypergeometric pFq"
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The generalized hypergeometric function ${}_pF_q$ is defined by | The generalized hypergeometric function ${}_pF_q$ is defined by | ||
| − | $${}_pF_q(a_1,a_2,\ldots,a_p;b_1,\ldots,b_q;z)=\displaystyle\ | + | $${}_pF_q(a_1,a_2,\ldots,a_p;b_1,\ldots,b_q;z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1)_k(a_2)_k\ldots(a_p)_k}{(b_1)_k(b_2)_k\ldots(b_q)_k} \dfrac{z^k}{k!},$$ |
where $(a_1)_k$ denotes the [[Pochhammer]] symbol. | where $(a_1)_k$ denotes the [[Pochhammer]] symbol. | ||
Revision as of 21:42, 17 June 2017
The generalized hypergeometric function ${}_pF_q$ is defined by $${}_pF_q(a_1,a_2,\ldots,a_p;b_1,\ldots,b_q;z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1)_k(a_2)_k\ldots(a_p)_k}{(b_1)_k(b_2)_k\ldots(b_q)_k} \dfrac{z^k}{k!},$$ where $(a_1)_k$ denotes the Pochhammer symbol.
Properties
Convergence of Hypergeometric pFq
Hypergeometric pFq terminates to a polynomial if an a_j is a nonpositive integer
Hypergeometric pFq diverges if a b_j is a nonpositive integer
Hypergeometric pFq converges for all z if p less than q+1
Hypergeometric pFq converges in the unit disk if p=q+1
Hypergeometric pFq diverges if p greater than q+1
Derivatives of Hypergeometric pFq
Differential equation for Hypergeometric pFq
Particular hypergeometric functions
Hypergeometric 0F0
Hypergeometric 1F0
Hypergeometric 0F1
Hypergeometric 1F1
Hypergeometric 2F1
Hypergeometric 1F2
Hypergeometric 2F0
Hypergeometric 2F1
Videos
Special functions - Hypergeometric series (9 March 2011)
External links
Notes on hypergeometric functions
Note on a hypergeometric series - Cayley
References
- 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $4.1 (1)$ (note: typo in the text, the sum there starts at $1$ but should start at $0$)
Hypergeometric ${}_pF_q$
