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
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__NOTOC__
$$\vec{a}^{\overline{k}} = \displaystyle\prod_{j=1}^p a_j^{\overline{k}},$$
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The generalized hypergeometric function ${}_pF_q$ is defined by
and
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$${}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1)_k(a_2)_k\ldots(a_p)_k}{(b_1)_k(b_2)_k\ldots(b_q)_k} \dfrac{z^k}{k!},$$
$$\vec{a}+k = \displaystyle\prod_{j=1}^p (a_j+k),$$
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where $(a_1)_k$ denotes the [[Pochhammer]] symbol.
(and similar for $\vec{b}^{\overline{k}}$).
 
Define the generalized hypergeometric function
 
$${}_pF_q(a_1,a_2,\ldots,a_p;b_1,\ldots,b_q;t)={}_pF_q(\vec{a};\vec{b};t)=\displaystyle\sum_{k=0}^{\infty}\dfrac{\displaystyle\prod_{j=1}^p a_j^{\overline{k}}}{\displaystyle\prod_{\ell=1}^q b_{\ell}^{\overline{k}}} \dfrac{t^k}{k!}.$$
 
  
=Convergence=
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=Properties=
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 />
 +
[[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.
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[[Derivatives of Hypergeometric pFq]]<br />
 +
[[Differential equation for Hypergeometric pFq]]<br />
  
The remaining convergence of the series can be split into three cases:
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=Videos=
 +
[https://www.youtube.com/watch?v=l8udH-Zb5Vs Special functions - Hypergeometric series (9 March 2011)]<br />
  
==Case I: $p<q+1$==
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=External links=
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[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 />
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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 />
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> Notice if $t=0$ then the series converges trivially, so suppose $t \neq 0$. We will apply the [[ratio_test | ratio test]]. Let $\alpha_k=\dfrac{\vec{a}^{\overline{k}}t^k}{\vec{b}^{\overline{k}}k!}$. Then
 
$$\begin{array}{ll}
 
L &= \displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{\alpha_{k+1}}{\alpha_k} \right| \\
 
&= \displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{\dfrac{\vec{a}^{\overline{k}}t^k}{\vec{b}^{\overline{k}}k!}}{\dfrac{\vec{a}^{\overline{k+1}}t^{k+1}}{\vec{b}^{\overline{k+1}}(k+1)!}} \right| \\
 
&= \displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{\vec{a}^{\overline{k}} \vec{b}^{\overline{k+1}}(k+1)!t^k }{\vec{b}^{\overline{k}} \vec{a}^{\overline{k+1}}k!t^{k+1}} \right| \\
 
&= \displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{k(\vec{b}+k)}{(\vec{a}+k)t} \right| \\
 
&= \displaystyle\lim_{k \rightarrow \infty} \left| \dfrac{O(k^{q+1})}{O(k^{p})}\right| \\
 
&= 0 < 1,
 
\end{array}$$
 
therefore the series converges for all $t \in \mathbb{C}$. █
 
</div></div>
 
  
==Case II: $p=q+1$==
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=References=
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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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 $0$)
<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}}: $5.1 (2)$
<div class="mw-collapsible-content">
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* {{BookReference|Special Functions of Mathematical Physics and Chemistry|1956|Ian N. Sneddon|prev=findme|next=findme}}: $\S 12 (12.4)$
<strong>Proof: █</strong>
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* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=findme|next=findme}}: $(9.1)$
</div></div>
 
  
==Case III: $p>q+1$==
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{{: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=
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[[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}
 
\dfrac{d^n}{dt^n} {}_pF_q(\vec{a};\vec{b};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>
 
<br />
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Proposition:</strong> Suppose that ${}_pF_q$ converges. Then
 
$$\dfrac{d^n}{dt^n} \left[ t^{\gamma} {}_pF_q(\vec{a};\vec{b};t) \right] = (\gamma-n+1)^{\overline{n}}t^{\gamma-n} {}_{p+1}F_{q+1}(\gamma+1,\vec{a};\gamma+1-n,\vec{b};t).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> First we suppose $n=0$ yielding the formula
 
$$\begin{array}{ll}
 
t^{\gamma}{}_pF_q(\vec{a};\vec{b};t) &= t^{\gamma-n} \displaystyle\sum_{k=0}^{\infty} \dfrac{\vec{a}}{\vec{b}} \dfrac{t^k}{k!} \\
 
&= t^{\gamma-n} \displaystyle\sum_{k=0}^{\infty} \dfrac{(\gamma+1)\vec{a}}{(\gamma+1)\vec{b}}\dfrac{t^k}{k!} \\
 
&= t^{\gamma-n} {}_{p+1}F_{q+1}(\gamma+1,\vec{a};\gamma+1-0,\vec{b};t),
 
\end{array}$$
 
obeying the formula. Now suppose that the formula is satisfied for $n=1,2,\ldots,N-1$. We will show now that the formula holds for $n=N$:
 
$$\begin{array}{ll}
 
\dfrac{d^{N}}{dt^{N}} \left[ t^{\gamma} {}_pF_q(\vec{a};\vec{b};t)\right] &= \dfrac{d}{dt} \left[ \dfrac{d^{N-1}}{dt^{N-1}} \left[ t^{\gamma} {}_pF_q(\vec{a};\vec{b};t) \right] \right] \\
 
&=\dfrac{d}{dt} \left[ (\gamma-(N-1)+1)^{\overline{N-1}} t^{\gamma-(N-1)} {}_{p+1}F_{q+1}(\gamma+1,\vec{a};\gamma+1-(N-1),\vec{b};t) \right] \\
 
&=(\gamma-N+2)^{\overline{N-1}}(\gamma-N+1)t^{\gamma-N}{}_{p+1}F_{q+1}(\gamma+1,\vec{a};\gamma-N+2,\vec{b};t) \\
 
&\hspace{4pt}+(\gamma-N+2)^{\overline{N-1}}t^{\gamma-N+1}\dfrac{(\gamma+1) \vec{a}}{(\gamma-N+2)\vec{b}} {}_{p+1}F_{q+1} (\gamma+2,\vec{a}+1;\gamma-N+3,\vec{b};t) \\
 
&= (\gamma-N+2)^{\overline{N-1}} \left\{ (\gamma-N+1)t^{\gamma-N}{}_{p+1}F_{q+1}(\gamma+1;\vec{a};\gamma-N+2,\vec{b};t)  \right. \\
 
&\hspace{4pt}+ \left. t^{\gamma-N+1} \dfrac{(\gamma+1)\vec{a}}{(\gamma-N+2)\vec{b}} {}_{p+1}F_{q+1}(\gamma+2,\vec{a}+1;\gamma-N+3,\vec{b};t) \right\} NEEDSWORK
 
\end{array}$$  █
 
</div></div>
 
 
 
=Differential equation=
 
Define the derivative operator $\vartheta=t \dfrac{d}{dt}$.Then
 
$$\vartheta t^k = t \dfrac{d}{dt} t^k = t(kt^{k-1})=kt^k.$$
 
 
 
<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] t^k \\
 
&= \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} \\
 
&= \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!} \\
 
&= \left[ t \displaystyle\prod_{i=1}^p (\vartheta + a_i) \right] y(t)
 
\end{array}$$
 
proves the claim. █
 
</div></div>
 
 
 
=Examples=
 
 
 
==${}_0F_0$==
 
#${}_0F_0(;;z)=e^z$
 
 
 
==${}_0F_1$==
 
#${}_0F_1 \left(;\dfrac{1}{2};-\dfrac{z^2}{4} \right)=\cos(z)$
 
#$z{}_0F_1 \left(;\dfrac{3}{2};-\dfrac{z^2}{4} \right)=\sin(z)$
 
 
 
==${}_1F_0$==
 
#${}_1F_0(-a;;z)=(1-z)^a$
 
 
 
==${}_1F_1$==
 
 
 
==${}_2F_0$==
 
{{:Bessel polynomial generalized hypergeometric}}
 
 
 
==${}_2F_1$==
 
{{:z2F1(1,1;2,-z) equals log(1+z)}}
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$2{}_2F_1(a,a+\frac{1}{2};\frac{1}{2};z)=(1+\sqrt{z})^{-2a}+(1-\sqrt{z})^{-2a}.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$e^{-az}=(2\cosh z)^{-a} \tanh(z) {}_2F_1(1+\frac{a}{2},\frac{1}{2}+\frac{a}{2};1+a;(\cosh z)^{-2}).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$\arcsin z = z {}_2F_1(\frac{1}{2},\frac{1}{2};\frac{3}{2};z^2).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$\arctan z = z {}_2F_1(\frac{1}{2},1;\frac{3}{2};-z^2).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$\log \left( \dfrac{1+z}{1-z} \right) = 2z {}_2F_1(\frac{1}{2},1;\frac{3}{2};z^2).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$T_n(x) = {}_2F_1 \left( -n,n ; \dfrac{1}{2}; \dfrac{1-x}{2} \right),$$
 
where $T_n$ denotes a [[Chebyshev T]] and ${}_2F_1$ denotes the [[hypergeometric pFq]].
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$U_n(x) = (n+1){}_2F_1 \left( -n,n+2 ; \dfrac{3}{2}; \dfrac{1-x}{2} \right),$$
 
where $U_n$ denotes a [[Chebyshev U]] and ${}_2F_1$ denotes the [[hypergeometric pFq]].
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$P_n(x) = {}_2F_1 \left( -n, n+2; \dfrac{3}{2}; \dfrac{1-x}{2} \right),$$
 
where $P_n$ denotes a [[Legendre polynomial]] and ${}_2F_1$ denotes the [[hypergeometric pFq]].
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
=Videos=
 
[https://www.youtube.com/watch?v=l8udH-Zb5Vs Special functions - Hypergeometric series]<br />
 
 
 
=References=
 
[http://www.johndcook.com/HypergeometricFunctions.pdf Notes on hypergeometric functions]<br />
 
Rainville's Special Functions<br />
 
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_555.htm Abramowitz and Stegun]
 

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 ${}_pF_q$