Revision as of 23:25, 3 March 2018

The generalized hypergeometric function ${}_pF_q$ is defined by $${}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1)_k(a_2)_k\ldots(a_p)_k}{(b_1)_k(b_2)_k\ldots(b_q)_k} \dfrac{z^k}{k!},$$ 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

Videos

Special functions - Hypergeometric series (9 March 2011)

External links

Notes on hypergeometric functions
Note on a hypergeometric series - Cayley

References

• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: 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 $0$)
• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $5.1 (2)$
• 1956: Ian N. Sneddon: Special Functions of Mathematical Physics and Chemistry ... (previous) ... (next): $\S 12 (12.4)$

Hypergeometric ${}_pF_q$

Hypergeometric 0F0

Hypergeometric 1F0

Hypergeometric 0F1

Hypergeometric 2F1