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Made the series representation more explicit by way of the gamma function. This way future readers don't have to guess what the exact representation of the generalized hypergeometric function looks like. |
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:<math>\begin{align}
(a)_0 &= 1, \\
(a)_n &= a(a+1)(a+2) \cdots (a+n-1) = \frac{\Gamma(a+n)}{\Gamma(a)}, && n \geq 1,
\end{align}</math>
where <math>\Gamma(x)</math> represents the [[Gamma function|gamma function]], this can be written
:<math>\,{}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdots(a_p)_n}{(b_1)_n\cdots(b_q)_n} \, \frac {z^n} {n!} = \frac{\Gamma(b_1)\cdots \Gamma(b_q)}{\Gamma(a_1)\cdots \Gamma(a_p)} \sum_{n=0}^{\infty} \frac{\Gamma(n+a_1)\cdots \Gamma(n+a_p)}{\Gamma(n+b_1)\cdots \Gamma(n+b_q)} \frac{z^n}{n!}.</math>
(Note that this use of the Pochhammer symbol is not standard; however it is the standard usage in this context.)
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:<math>\int_0^x\sqrt{1+y^\alpha}\,\mathrm{d}y=\frac{x}{2+\alpha}\left \{\alpha\;{}_2F_1\left(\tfrac{1}{\alpha},\tfrac{1}{2};1+\tfrac{1}{\alpha};-x^\alpha \right) +2\sqrt{x^\alpha+1} \right \},\qquad \alpha\neq0.</math>
===The series <sub>2</sub>''F''<sub>2</sub>===
The hypergeometric series <math>{}_2F_2</math> is generally associated with integrals of products of power functions and the exponential function. As such, the [[Exponential integral|exponential integral]] can be written as:
:<math>\operatorname{Ei}(x)=x{}_2F_2(1,1;2,2;x)+\ln x+\gamma.</math>
===The series <sub>3</sub>''F''<sub>0</sub>===
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</ref>
::<math>\operatorname{BR}(a) = -a \; {}_4F_3\left( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} ; \frac{1}{2}, \frac{3}{4}, \frac{5}{4} ; \frac{3125a^4}{256} \right).</math>
The partition function <math>Z(K)</math> of the 2D isotropic [[Square lattice Ising model|Ising model]] with no external magnetic field was found by [[Lars Onsager|Onsager]] in the 1940s and can be expressed as<ref name="Ising">
{{cite journal
| last = Viswanathan | first = G. M.
| year = 2014
| title = The hypergeometric series for the partition function of the 2-D Ising model
| journal = Journal of Statistical Mechanics: Theory and Experiment
| volume = 2015
| issue = 7
| page = 07004
| doi = 10.1088/1742-5468/2015/07/P07004
| arxiv = 1411.2495
| bibcode = 2015JSMTE..07..004V
}}
</ref>
::<math>\ln Z(K) = \ln(2 \cosh 2K) -k^2 {}_4F_3\left( 1, 1, \frac{3}{2}, \frac{3}{2} ; 2, 2, 2; 16k^2 \right),</math>
with <math>K=\frac{J}{k_\mathrm{B}T}</math> and <math>k=\frac{1}{2}\tanh 2K\,\operatorname{sech} 2K</math>.
===The series <sub>q+1</sub>''F''<sub>q</sub>===
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* {{cite book | last= Bailey | first= W.N. | title= Generalized Hypergeometric Series | publisher= Cambridge University Press | ___location= London | year= 1935 | series= Cambridge Tracts in Mathematics and Mathematical Physics | volume= 32 | zbl= 0011.02303 }}
* {{cite journal | last= Dixon | first= A.C. | title= Summation of a certain series | journal= Proc. London Math. Soc. | year= 1902 | volume= 35 | issue= 1 | pages= 284–291 | doi=10.1112/plms/s1-35.1.284 | jfm= 34.0490.02 | url= https://zenodo.org/record/1433433 }}
* {{cite journal | last= Dougall | first= J. | title= On Vandermonde's theorem and some more general expansions | journal= Proc. Edinburgh Math. Soc. | year= 1907 | volume= 25 | pages= 114–132 | doi= 10.1017/S0013091500033642
*{{Cite book | last1=Erdélyi | first1=Arthur | last2=Magnus | first2=Wilhelm | author2-link=Wilhelm Magnus | last3=Oberhettinger | first3=Fritz | last4=Tricomi | first4=Francesco G. | title=Higher transcendental functions. Vol. III | publisher=McGraw-Hill Book Company, Inc., New York-Toronto-London | mr=0066496 | year=1955}}
* {{cite book | last1= Gasper | first1= George | last2= Rahman | first2= Mizan | author-link2= Mizan Rahman | title= Basic Hypergeometric Series | edition= 2nd | year= 2004 | series= Encyclopedia of Mathematics and Its Applications | volume= 96 | publisher= Cambridge University Press | ___location= Cambridge, UK | mr= 2128719 | zbl= 1129.33005 | isbn= 978-0-521-83357-8 }} (the first edition has {{isbn|0-521-35049-2}})
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