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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
| + | __NOTOC__ |
| − | $$\vec{a}^{\overline{k}} = \displaystyle\prod_{j=1}^p a_j^{\overline{k}},$$ | + | The generalized hypergeometric function ${}_pF_q$ is defined by |
| − | and
| + | $${}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1)_k(a_2)_k\ldots(a_p)_k}{(b_1)_k(b_2)_k\ldots(b_q)_k} \dfrac{z^k}{k!},$$ |
| − | $$\vec{a}+k = \displaystyle\prod_{j=1}^p (a_j+k),$$
| + | where $(a_1)_k$ denotes the [[Pochhammer]] symbol. |
| − | (and similar for $\vec{b}^{\overline{k}}$).
| |
| − | Define the generalized hypergeometric function
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| − | $${}_pF_q(a_1,a_2,\ldots,a_p;b_1,\ldots,b_q;t)={}_pF_q(\vec{a};\vec{b};t)=\displaystyle\sum_{k=0}^{\infty}\dfrac{\displaystyle\prod_{j=1}^p a_j^{\overline{k}}}{\displaystyle\prod_{\ell=1}^q b_{\ell}^{\overline{k}}} \dfrac{t^k}{k!}.$$ | |
| | | | |
| − | ==Convergence== | + | =Properties= |
| − | If any of the $a_j$'s is a a nonpositive integer, then the series terminates and is a polynomial.
| + | [[Convergence of Hypergeometric pFq]]<br /> |
| | + | [[Hypergeometric pFq terminates to a polynomial if an a_j is a nonpositive integer]]<br /> |
| | + | [[Hypergeometric pFq diverges if a b_j is a nonpositive integer]]<br /> |
| | + | [[Hypergeometric pFq converges for all z if p less than q+1]]<br /> |
| | + | [[Hypergeometric pFq converges in the unit disk if p=q+1]]<br /> |
| | + | [[Hypergeometric pFq diverges if p greater than q+1]]<br /> |
| | | | |
| − | If any of the $b_{\ell}$'s is a nonpositive integer, the series diverges because of divison by zero.
| + | [[Derivatives of Hypergeometric pFq]]<br /> |
| | + | [[Differential equation for Hypergeometric pFq]]<br /> |
| | | | |
| − | The remaining convergence of the series can be split into three cases:
| + | =Videos= |
| | + | [https://www.youtube.com/watch?v=l8udH-Zb5Vs Special functions - Hypergeometric series (9 March 2011)]<br /> |
| | | | |
| − | ===Case I: $p<q+1$=== | + | =External links= |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| + | [http://www.johndcook.com/HypergeometricFunctions.pdf Notes on hypergeometric functions]<br /> |
| − | <strong>Proposition:</strong> The series ${}_pF_q$ converges for all $t \in \mathbb{C}$.<br />
| + | [http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0016%7CLOG_0038 Note on a hypergeometric series - Cayley]<br /> |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof: █</strong>
| |
| − | </div></div>
| |
| | | | |
| − | ===Case II: $p=q+1$=== | + | =References= |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=findme|next=Pochhammer}}: $4.1 (1)$ (note: typo in the text, the sum there starts at $1$ but should start at $0$) |
| − | <strong>Proposition:</strong> The series ${}_pF_q$ converges for all $t\in \mathbb{C}$ with $|t|<1$.<br />
| + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=findme|next=Pochhammer}}: $5.1 (2)$ |
| − | <div class="mw-collapsible-content">
| + | * {{BookReference|Special Functions of Mathematical Physics and Chemistry|1956|Ian N. Sneddon|prev=findme|next=findme}}: $\S 12 (12.4)$ |
| − | <strong>Proof: █</strong>
| + | * {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=findme|next=findme}}: $(9.1)$ |
| − | </div></div>
| |
| | | | |
| − | ===Case III: $p>q+1$===
| + | {{:Hypergeometric functions footer}} |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| |
| − | <strong>Proposition:</strong> The series ${}_pF_q$ diverges for all $t \in \mathbb{C}$.<br />
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof: █</strong>
| |
| − | </div></div>
| |
| | | | |
| − | ==Derivatives==
| + | [[Category:SpecialFunction]] |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| |
| − | <strong>Proposition:</strong> Suppose that ${}_pF_q$ converges. Then
| |
| − | $$\dfrac{d^n}{dt^n} {}_pF_q(\vec{a};\vec{b};t)=\dfrac{\vec{a}^{\overline{n}}}{\vec{b}^{\overline{n}}} {}_pF_q(\vec{a}+n;\vec{b}+n;t).$$
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong> The computation
| |
| − | $$\begin{array}{ll}
| |
| − | \Delta^{n} {}_pF_q(\vec{a}_p;\vec{b}_q;t) &= \dfrac{d^n}{dt^n}\displaystyle\sum_{k=0}^{\infty} \dfrac{ \vec{a}^{\overline{k}} }{ \vec{b}^{\overline{k}} } \dfrac{t^{\underline{k}}}{k!} \\
| |
| − | &= \displaystyle\sum_{k=n}^{\infty} \dfrac{ \vec{a}^{\overline{k}} }{ \vec{b}^{\overline{k}} } \dfrac{t^{\underline{k-n}}}{(k-n)!} \\
| |
| − | &= \displaystyle\sum_{k=0}^{\infty} \dfrac{ \vec{a}^{\overline{k+n}} }{ \vec{b}^{\overline{k+n}} } \dfrac{t^{\underline{k}}}{k!} \\
| |
| − | &=\dfrac{ \vec{a}^{\overline{n}} }{ \vec{b}^{\overline{n}} } \displaystyle\sum_{k=0}^{\infty} \dfrac{ (\vec{a}+n)^{\overline{k}} }{ (\vec{b}+n)^{\overline{k}} } \dfrac{t^{\underline{k}}}{k!} \\
| |
| − | &=\dfrac{ \vec{a}^{\overline{n}} }{ \vec{b}^{\overline{n}} } {}_pF_q(\vec{a}+n;\vec{b}+n;t)
| |
| − | \end{array}$$
| |
| − | proves the claim. █
| |
| − | </div></div>
| |
| − | | |
| − | ==Differential equation==
| |
| − | Define the derivative operator $\vartheta=t \dfrac{d}{dt}$.Then
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| − | $$\vartheta t^k = t \dfrac{d}{dt} t^k = t(kt^{k-1})=kt^k.$$
| |
| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| |
| − | <strong>Proposition:</strong> The operator $\vartheta$ is a [[Linear_operator | linear operator]]. <br />
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof: █</strong>
| |
| − | </div></div>
| |
| − | <br />
| |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| |
| − | <strong>Theorem:</strong> Define $y(t)={}_pF_q(\vec{a};\vec{b};t)$. Then $y$ satisfies
| |
| − | $$(\dagger) \hspace{35pt} \left[ \vartheta \displaystyle\prod_{j=1}^q (\vartheta + b_j-1) - t \displaystyle\prod_{i=1}^p (\vartheta+a_i) \right]y=0.$$
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong>
| |
| − | First compute
| |
| − | $$\begin{array}{ll}
| |
| − | \left[ t \displaystyle\prod_{i=1}^p (\vartheta+a_i) \right] y(t) &= \left[ t \displaystyle\prod_{i=1}^p (\vartheta + a_i) \right] \displaystyle\sum_{k=0}^{\infty} \dfrac{\vec{a}^{\overline{k}}}{\vec{b}^{\overline{k}}} \dfrac{t^k}{k!} \\
| |
| − | &= t\displaystyle\sum_{k=0}^{\infty} \dfrac{\vec{a}^{\overline{k}}}{\vec{b}^{\overline{k}}} \left[ \displaystyle\prod_{i=1}^p (\vartheta + a_i) \right] \dfrac{t^k}{k!} \\
| |
| − | &= t\displaystyle\sum_{k=0}^{\infty} \dfrac{\vec{a}^{\overline{k}}}{\vec{b}^{\overline{k}}k!} \left[ \displaystyle\prod_{i=1}^p (\vartheta + a_i) \right] t^k \\
| |
| − | &= t\displaystyle\sum_{k=0}^{\infty} \dfrac{\vec{a}^{\overline{k}}}{\vec{b}^{\overline{k}}} \left[ \displaystyle\prod_{i=1}^p (k+a_i) \right] \dfrac{t^k}{k!} \\
| |
| − | &=t\displaystyle\sum_{k=0}^{\infty} \dfrac{(\vec{a}+k)\vec{a}^{\overline{k}}}{\vec{b}^{\overline{k}}} \dfrac{t^k}{k!}. \\
| |
| − | \end{array}$$
| |
| − | Now the computation
| |
| − | $$\begin{array}{ll}
| |
| − | \left[ \vartheta \displaystyle\prod_{j=1}^q (\vartheta + b_j -1) \right]y(t) &= \left[ \vartheta \displaystyle\prod_{j=1}^q (\vartheta+b_j-1) \right]\displaystyle\sum_{k=0}^{\infty} \dfrac{\vec{a}^{\overline{k}}}{\vec{b}^{\overline{k}}} \dfrac{t^k}{k!} \\
| |
| − | &=\displaystyle\sum_{k=0}^{\infty} \dfrac{\vec{a}^{\overline{k}}}{\vec{b}^{\overline{k}}k!} \left[ \vartheta \displaystyle\prod_{j=1}^q (\vartheta + b_j -1) \right] t^k \\
| |
| − | &= \displaystyle\sum_{k=1}^{\infty} \dfrac{\vec{a}^{\overline{k}}}{k!} \left[ \dfrac{\displaystyle\prod_{j=1}^q (k + b_j -1)}{b^{\overline{k}}} \right] \vartheta t_k \\
| |
| − | &= \displaystyle\sum_{k=1}^{\infty} \dfrac{\vec{a}^{\overline{k}}}{k!} \left[ k\displaystyle\prod_{j=1}^q \dfrac{k+b_j-1}{b_j(b_j+1)\ldots(b_j+k-1)} \right] \\
| |
| − | &= \displaystyle\sum_{k=1}^{\infty} \dfrac{\vec{a}^{\overline{k}}}{k!} \left[ \displaystyle\prod_{j=1}^q \dfrac{1}{b_j(b_j+1)\ldots(b_j+k-2)} \right] t^k \\
| |
| − | &= \displaystyle\sum_{k=1}^{\infty} \dfrac{\vec{a}^{\overline{k}}}{\vec{b}^{\overline{k-1}}(k-1)!} t^k \\
| |
| − | &= \displaystyle\sum_{k=0}^{\infty} \dfrac{\vec{a}^{\overline{k+1}}}{\vec{b}^{\overline{k}}k!}t^{k+1} \\
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| − | &= \displaystyle\sum_{k=0}^{\infty} \dfrac{(\vec{a}+k)\vec{a}^{\overline{k}}}{\vec{b}^{\overline{k}}} \dfrac{t^{k+1}}{k!} \\
| |
| − | &= t\displaystyle\sum_{k=0}^{\infty} \dfrac{(\vec{a}+k)\vec{a}^{\overline{k}}}{\vec{b}^{\overline{k}}} \dfrac{t^k}{k!} \\
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| − | &= \left[ t \displaystyle\prod_{i=1}^p (\vartheta + a_i) \right] y(t)
| |
| − | \end{array}$$
| |
| − | proves the claim. █
| |
| − | </div></div>
| |
| − | | |
| − | ==References==
| |
| − | Rainville's Special Functions
| |
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.
