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 $\vec{a}=(a_1,\ldots,a_p)$ and $\vec{b}=(b_1,\ldots,b_q)$ and we define the notations
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__NOTOC__
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$$\vec{a}^{\overline{k}} = \displaystyle\prod_{j=1}^p a_j^{\overline{k}},$$
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The generalized hypergeometric function ${}_pF_q$ is defined by
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and
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$${}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1)_k(a_2)_k\ldots(a_p)_k}{(b_1)_k(b_2)_k\ldots(b_q)_k} \dfrac{z^k}{k!},$$
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$$\vec{a}+k = \displaystyle\prod_{j=1}^p (a_j+k),$$
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where $(a_1)_k$ denotes the [[Pochhammer]] symbol.
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(and similar for $\vec{b}^{\overline{k}}$).
 
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Define the generalized hypergeometric function
 
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$${}_pF_q(a_1,a_2,\ldots,a_p;b_1,\ldots,b_q;t)={}_pF_q(\vec{a};\vec{b};t)=\displaystyle\sum_{k=0}^{\infty}\dfrac{\displaystyle\prod_{j=1}^p a_j^{\overline{k}}}{\displaystyle\prod_{\ell=1}^q b_{\ell}^{\overline{k}}} \dfrac{t^k}{k!}.$$
 
  
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==Convergence==
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=Properties=
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If any of the $a_j$'s is a a nonpositive integer, then the series terminates and is a polynomial.
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[[Convergence of Hypergeometric pFq]]<br />
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[[Hypergeometric pFq terminates to a polynomial if an a_j is a nonpositive integer]]<br />
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[[Hypergeometric pFq diverges if a b_j is a nonpositive integer]]<br />
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[[Hypergeometric pFq converges for all z if p less than q+1]]<br />
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[[Hypergeometric pFq converges in the unit disk if p=q+1]]<br />
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[[Hypergeometric pFq diverges if p greater than q+1]]<br />
  
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If any of the $b_{\ell}$'s is a nonpositive integer, the series diverges because of divison by zero.
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[[Derivatives of Hypergeometric pFq]]<br />
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[[Differential equation for Hypergeometric pFq]]<br />
  
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The remaining convergence of the series can be split into three cases:
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=Videos=
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[https://www.youtube.com/watch?v=l8udH-Zb5Vs Special functions - Hypergeometric series (9 March 2011)]<br />
  
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===Case I: $p<q+1$===
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=External links=
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<strong>Proposition:</strong> The series ${}_pF_q$ converges for all $t \in \mathbb{C}$.<br />
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[http://www.johndcook.com/HypergeometricFunctions.pdf Notes on hypergeometric functions]<br />
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<strong>Proof: █</strong>  
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[http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0016%7CLOG_0038 Note on a hypergeometric series - Cayley]<br />
  
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===Case II: $p=q+1$===
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=References=
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<strong>Proposition:</strong> The series ${}_pF_q$ converges for all $t\in \mathbb{C}$ with $|t|<1$.<br />
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* {{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 $0$)
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<strong>Proof: █</strong>
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* {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=findme|next=Pochhammer}}: $5.1 (2)$
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* {{BookReference|Special Functions of Mathematical Physics and Chemistry|1956|Ian N. Sneddon|prev=findme|next=findme}}: $\S 12 (12.4)$
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* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=findme|next=findme}}: $(9.1)$
  
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===Case III: $p>q+1$===
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{{:Hypergeometric functions footer}}
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<strong>Proposition:</strong> The series ${}_pF_q$ diverges for all $t \in \mathbb{C}$.<br />
 
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<strong>Proof: █</strong>
 
  
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==Differential equation==
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[[Category:SpecialFunction]]
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Define the derivative operator $\vartheta=t \dfrac{d}{dt}$.Then
 
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$$\vartheta t^k = t \dfrac{d}{dt} t^k = t(kt^{k-1})=kt^k.$$
 
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<strong>Proposition:</strong> The operator $\vartheta$ is a [[Linear_operator | linear operator]]. <br />
 
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<strong>Proof: █</strong>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Theorem:</strong> Define $y(t)={}_pF_q(\vec{a};\vec{b};t)$. Then $y$ satisfies
 
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$$(\dagger) \hspace{35pt} \left[ \vartheta \displaystyle\prod_{j=1}^q (\vartheta + b_j-1) - t \displaystyle\prod_{i=1}^p (\vartheta+a_i) \right]y=0.$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong>
 
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First compute
 
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$$\begin{array}{ll}
 
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\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!} \\
 
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&= 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!} \\
 
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&= 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 \\
 
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&= 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!} \\
 
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&=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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\end{array}$$
 
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Now the computation
 
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$$\begin{array}{ll}
 
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\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!} \\
 
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&=\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 \\
 
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&= \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 \\
 
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&= \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] \\
 
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&= \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 \\
 
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&= \displaystyle\sum_{k=1}^{\infty} \dfrac{\vec{a}^{\overline{k}}}{\vec{b}^{\overline{k-1}}(k-1)!} t^k \\
 
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&= \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!} \\
 
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&= 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)
 
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\end{array}$$
 
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proves the claim. █
 
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</div></div>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Lemma:</strong> If $p \leq q$, then equation $(\dagger)$ reduces to
 
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$$t^q \dfrac{d^{q+1}y}{dt^{q+1}}(t) + \displaystyle\sum_{j=1}^q z^{j-1}(\alpha_jz+\beta_j)\dfrac{d^jy}{dt^j}+\alpha_0 w =0.$$
 
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</div>
 
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==References==
 
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Rainville's Special Functions
 

Latest revision as of 14:42, 15 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

Hypergeometric functions
Hypergeometricthumb.png
Hypergeometric ${}_pF_q$


Hypergeometric 0F0

Hypergeometric 1F0

Hypergeometric 0F1

Hypergeometric 2F1
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