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Gosper's algorithm
In mathematics, Gosper's algorithm is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose we have ''a''(1) + ... + ''a''(''n'') = ''S''(''n'') − ''S''(0), where ''S''(''n'') is a hypergeometric term (i.e., ''S''(''n'' + 1)/''S''(''n'') is a rational function of ''n''); then necessarily ''a''(''n'') is itself a hypergeometric term, and given the formula for ''a''(''n'') Gosper's algorithm finds that for ''S(''n'').
==Outline of the algorithm==
Step 1: Find a polynomial ''p'' such that, writing ''b''(''n'') = ''a''(''n'')/''p''(''n''), the ratio ''b''(''n'')/''b''(''n'' − 1) has the form ''q''(''n'')/''r''(''n'') where ''q'' and ''r'' are polynomials and no ''q''(''n'') has a nontrivial factor with ''r''(''n'' + ''j'') for ''j'' = 0, 1, 2, ... . (This is always possible, whether or not the series is summable in closed form.)
Step 2: Find a polynomial ''ƒ'' such that ''S''(''n'') = ''q''(''n'' + 1)/''p''(''n'') ''ƒ''(''n'') ''a''(''n''). If the series is summable in closed form then clearly a rational function ''ƒ'' with this property exists; in fact it must always be a polynomial, and an upper bound on its degree can be found. Determining ''ƒ'' (or finding that there is no such ''ƒ'') is then a matter of solving a system of linear equations.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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