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In mathematics, a bilateral hypergeometric series is a series Σ''a''''n'' summed over ''all'' integers ''n'', and such that the ratio :''a''''n''/''a''''n''+1 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''. 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. ==Definition== The bilateral hypergeometric series ''p''H''p'' is defined by : where : is the rising factorial or Pochhammer symbol. Usually the variable ''z'' is taken to be 1, in which case it is omitted from the notation. It is possible to define the series ''p''H''q'' 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「bilateral hypergeometric series」の詳細全文を読む スポンサード リンク
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