# 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?

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?

Some examples where having integral in elementary functions results in having indefinite sum in elementary functions plus generalized polygamma:

http://en.wikipedia.org/wiki/User:MathFacts/Alternative

Under generalized polygamma I mean the generalization of Polygamma following this paper: http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf It differs from the polygamma used in Mathematica by being balanced.

This generalization allows to express Zeta function and the Bernoulli polynomials in terms of generalyzed polygamma function: http://en.wikipedia.org/wiki/Generalized_polygamma_function

Well probably this question should be confined to the continuous non-periodic functions.

• I don't know. The antiderivative of the tangent function is elementary, and we had a question here recently on what the sum of the tangent function might look like. – Gerry Myerson Oct 17 '10 at 10:29
• You can ignore my comment above, as I see you already know about that other recent question. You might have mentioned that and saved me some work. – Gerry Myerson Oct 17 '10 at 10:32
• It's great that you found the answer to mathoverflow.net/questions/41011/… after everyone basically thought it couldn't be done. I hope the currently accepted answer is unaccepted and yours is accepted and everyone votes it up. – decomwe Oct 17 '10 at 14:53