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| In mathematics, a '''bilateral hypergeometric series''' is a series Σ''a''<sub>''n''</sub> summed over ''all'' integers ''n'', and such that the ratio
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| :''a''<sub>''n''</sub>/''a''<sub>''n''+1</sub>
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| of two terms is a rational function of ''n''. The definition of the [[generalized hypergeometric series]] is similar, except that the terms with negative ''n'' must vanish; the bilateral series will in general have infinite numbers of non-zero terms for both positive and negative ''n''.
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| The bilateral hypergeometric series fails to converge for most rational functions, though it can be analytically continued to a function defined for most rational functions. There are several summation formulas giving its values for special values where it does converge.
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| ==Definition==
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| The bilateral hypergeometric series <sub>''p''</sub>H<sub>''p''</sub> is defined by
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| :<math>{}_pH_p(a_1,\ldots,a_p;b_1,\ldots,b_p;z)=
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| {}_pH_p\left(\begin{matrix}a_1&\ldots&a_p\\b_1&\ldots&b_p\\ \end{matrix};z\right)=
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| \sum_{n=-\infty}^\infty
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| \frac{(a_1)_n(a_2)_n\ldots(a_p)_n}{(b_1)_n(b_2)_n\ldots(b_p)_n}z^n</math>
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| where
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| :<math>(a)_n=a(a+1)(a+2)\cdots(a+n-1)\,</math>
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| is the [[rising factorial]] or [[Pochhammer symbol]].
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| Usually the variable ''z'' is taken to be 1, in which case it is omitted from the notation.
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| It is possible to define the series <sub>''p''</sub>H<sub>''q''</sub> with different ''p'' and ''q'' in a similar way, but this either fails to converge or can be reduced to the usual hypergeomtric series by changes of variables.
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| ==Convergence and analytic continuation==
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| Suppose that none of the variables ''a'' or ''b'' are integers, so that all the terms of the series are finite and non-zero. Then the terms with ''n''<0 diverge if |''z''| <1, and the terms with ''n''>0 diverge if |''z''| >1, so the series cannot converge unless |''z''|=1. When |''z''|=1, the series converges if
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| :<math>\Re(b_1+\cdots b_n -a_1-\cdots - a_n) >1. </math>
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| The bilateral hypergeometric series can be analytically continued to a multivalued meromorphic function of several variables whose singularities are
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| branch points at ''z'' = 0 and ''z''=1 and simple poles at ''a''<sub>''i''</sub> = −1, −2,... and ''b''<sub>''i''</sub> = 0, 1, 2, ...
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| This can be done as follows. Suppose that none of the ''a'' or ''b'' variables are integers. The terms with ''n'' positive converge for |''z''| <1 to a function satisfying an inhomogeneous linear equation with singularities at ''z'' = 0 and ''z''=1, so can be continued to a multivalued function with these points as branch points. Similarly the terms with ''n'' negative converge for |''z''| >1 to a function satisfying an inhomogeneous linear equation with singularities at ''z'' = 0 and ''z''=1, so can also be continued to a multivalued function with these points as branch points. The sum of these functions gives the analytic continuation of the bilateral hypergeometric series to all values of ''z'' other than 0 and 1, and satisfies a [[linear differential equation]] in ''z'' similar to the hypergeometric differential equation.
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| ==Summation formulas==
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| ===Dougall's bilateral sum===
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| :<math> {}_2H_2(a,b;c,d;1)= \sum_{-\infty}^\infty\frac{(a)_n(b)_n}{(c)_n(d)_n}= \frac{\Gamma(d)\Gamma(c)\Gamma(1-a)\Gamma(1-b)\Gamma(c+d-a-b-1)}{\Gamma(c-a)\Gamma(c-b)\Gamma(d-a)\Gamma(d-b)} </math> | |
| {{harv|Dougall|1907}}
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| This is sometimes written in the equivalent form
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| :<math>\sum_{n=-\infty}^\infty
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| \frac {\Gamma(a+n) \Gamma(b+n)}{\Gamma(c+n)\Gamma(d+n)} =
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| \frac {\pi^2}{\sin (\pi a) \sin (\pi b)}
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| \frac {\Gamma (c+d-a-b-1)}{\Gamma(c-a) \Gamma(d-a) \Gamma(c-b) \Gamma(d-b)}.</math>
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| ===Bailey's formula===
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| {{harv|Bailey|1959}} gave the following generalization of Dougall's formula:
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| :<math> {}_3H_3(a,b, f+1;d,e,f;1)=
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| \sum_{-\infty}^\infty\frac{(a)_n(b)_n(f+1)_n}{(d)_n(e)_n(f)_n}= \lambda\frac{\Gamma(d)\Gamma(e)\Gamma(1-a)\Gamma(1-b)\Gamma(d+e-a-b-2)}{\Gamma(d-a)\Gamma(d-b)\Gamma(e-a)\Gamma(e-b)} </math>
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| where
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| :<math> \lambda=f^{-1}\left[(f-a)(f-b)-(1+f-d)(1+f-e)\right].</math>
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| ==See also==
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| *[[basic bilateral hypergeometric series]]
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| ==References==
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| * {{citation | authorlink= W. N. Bailey | last1= Bailey | first1= W. N. | title= On the sum of a particular bilateral hypergeometric series <sub>3</sub>''H''<sub>3</sub> | journal= The Quarterly Journal of Mathematics. Oxford. Second Series | volume= 10 | year= 1959 | pages= 92–94 | issn= 0033-5606 | doi= 10.1093/qmath/10.1.92 | mr= 0107727 }}
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| * {{citation | last= Dougall | first= J. |authorlink=John Dougall (mathematician)| title= On Vandermonde's Theorem and Some More General Expansions | journal= Proc. Edinburgh Math. Soc. | volume= 25 | year= 1907 | pages= 114–132 | doi= 10.1017/S0013091500033642}}
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| * {{citation | last1= Slater | first1= Lucy Joan | authorlink= Lucy Joan Slater | title= Generalized hypergeometric functions | location= Cambridge, UK | publisher= Cambridge University Press | isbn= 0-521-06483-X | mr= 0201688 | year= 1966}} (there is a 2008 paperback with ISBN 978-0-521-09061-2)
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| [[Category:Hypergeometric functions]]
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My namе is Lucia (23 years old) and my hobbіes are Microscopy and Disc golf.
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