Gosper's algorithm: Difference between revisions

Revision as of 07:22, 6 February 2024

Latest revision as of 17:01, 8 June 2025

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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''&nbsp;+&nbsp;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.<ref>{{cite book |author-first1last1=MarkoPetkovšek |author-last1first1=PetkovšekMarko |author-link1=Marko Petkovšek |author-first2url=Herberthttp://www.math.upenn.edu/~wilf/AeqB.html |title=A&nbsp;=&nbsp;B |author-last2=Wilf |author-first2=Herbert |author-link2=Herbert Wilf |author-first3last3=DoronZeilberger |author-last3first3=ZeilbergerDoron |author-link3=Doron Zeilberger |titledate=A&nbsp;=&nbsp;B1996 |publisher=[[A K Peters Ltd.]] |date=1996 |isbn=1-56881-063-6 |url=http://www.math.upenn.edu/~wilf/AeqB.html |access-date=2020-01-10 |url-status=live |archive-url=https://web.archive.org/web/20190711145429/https://www.math.upenn.edu/~wilf/AeqB.html |archive-date=2019-07-11 |url-status=live}} [https://web.archive.org/web/20190726143843/https://www.math.upenn.edu/~wilf/AeqB.pdf] [https://web.archive.org/web/20200111042729/https://sites.math.rutgers.edu/%7Ezeilberg/AeqB.pdf] [https://web.archive.org/web/20170329070514/http://www.fmf.uni-lj.si/aeqb/AeqB.pdf]</ref>

==Relationship to Wilf&ndash;Zeilberger pairs==

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==Definite versus indefinite summation==

Gosper's algorithm finds (where possible) a hypergeometric closed form for the ''indefinite'' sum of hypergeometric terms. It can happen that there is no such closed form, but that the sum over ''all'' ''n'', or some particular set of values of ''n'', has a closed form. This question is only meaningful when the coefficients are themselves functions of some other variable. So, suppose ''a''(''n'',''k'') is a hypergeometric term in both ''n'' and ''k'': that is, ''a''(''n'',&nbsp;''k'')/''a''(''n''&nbsp;&minus;&nbsp;1,''k'') and ''a''(''n'',&nbsp;''k'')/''a''(''n'',&nbsp;''k''&nbsp;&minus;&nbsp;1) are rational functions of ''n'' and ''k''. Then [[Zeilberger's algorithm]] and [[Petkovšek's algorithm]] may be used to find closed forms for the sum over ''k'' of ''a''(''n'',&nbsp;''k'').

==History==