Difference between revisions of "Hypergeometric pFq"
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Let $p,q \in \{0,1,2,\ldots\}$ and $a_j,b_{\ell} \in \mathbb{R}$ for $j=1,\ldots,p$ and $\ell=1,\ldots,q$. We will use the notation $\vec{a}=\displaystyle\prod_{j=1}^p a_j$ and $\vec{b}=\displaystyle\prod_{\ell=1}^q b_{\ell}$ and we define the notations | Let $p,q \in \{0,1,2,\ldots\}$ and $a_j,b_{\ell} \in \mathbb{R}$ for $j=1,\ldots,p$ and $\ell=1,\ldots,q$. We will use the notation $\vec{a}=\displaystyle\prod_{j=1}^p a_j$ and $\vec{b}=\displaystyle\prod_{\ell=1}^q b_{\ell}$ and we define the notations | ||
$$\vec{a}^{\overline{k}} = \displaystyle\prod_{j=1}^p a_j^{\overline{k}},$$ | $$\vec{a}^{\overline{k}} = \displaystyle\prod_{j=1}^p a_j^{\overline{k}},$$ | ||
Revision as of 05:23, 5 July 2016
Let $p,q \in \{0,1,2,\ldots\}$ and $a_j,b_{\ell} \in \mathbb{R}$ for $j=1,\ldots,p$ and $\ell=1,\ldots,q$. We will use the notation $\vec{a}=\displaystyle\prod_{j=1}^p a_j$ and $\vec{b}=\displaystyle\prod_{\ell=1}^q b_{\ell}$ and we define the notations $$\vec{a}^{\overline{k}} = \displaystyle\prod_{j=1}^p a_j^{\overline{k}},$$ and $$\vec{a}+k = \displaystyle\prod_{j=1}^p (a_j+k),$$ (and similar for $\vec{b}^{\overline{k}}$). Define the generalized hypergeometric function $${}_pF_q(a_1,a_2,\ldots,a_p;b_1,\ldots,b_q;t)={}_pF_q(\vec{a};\vec{b};t)=\displaystyle\sum_{k=0}^{\infty}\dfrac{\displaystyle\prod_{j=1}^p a_j^{\overline{k}}}{\displaystyle\prod_{\ell=1}^q b_{\ell}^{\overline{k}}} \dfrac{t^k}{k!}.$$
Properties
Convergence of Hypergeometric pFq
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
References
Notes on hypergeometric functions
Rainville's Special Functions
Abramowitz and Stegun
Note on a hypergeometric series - Cayley
