Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T04:57:18Z http://mathoverflow.net/feeds/question/42484 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42484/can-any-antidifference-indefinite-sum-of-a-function-be-expressed-in-elementary Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions? Anixx 2010-10-17T10:23:00Z 2010-10-17T10:32:51Z <p>If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be expressed in elementary functions plus generalized polygamma function?</p> <p>Some examples where having integral in elementary functions results in having indefinite sum in elementary functions plus generalized polygamma:</p> <p><a href="http://en.wikipedia.org/wiki/User:MathFacts/Alternative" rel="nofollow">http://en.wikipedia.org/wiki/User:MathFacts/Alternative</a></p> <p>Under generalized polygamma I mean the generalization of Polygamma following this paper: <a href="http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf" rel="nofollow">http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf</a> It differs from the polygamma used in Mathematica by being <em>balanced</em>.</p> <p>This generalization allows to express Zeta function and the Bernoulli polynomials in terms of generalyzed polygamma function: <a href="http://en.wikipedia.org/wiki/Generalized_polygamma_function" rel="nofollow">http://en.wikipedia.org/wiki/Generalized_polygamma_function</a></p> <p>Well probably this question should be confined to the continuous non-periodic functions.</p>