I am sorry if the answer to my question is well-known. I am quite new in this topic, so it will be also nice to have a reference, if it exists.


I was wondered if there exists a nice closed formula for a logarithm of an arbitrary hypergeometric series in the terms of, say, a linear combination of some other hypergeometric series. The reason that makes me believe in the existence of such a formula is the following.


It is an easy exercise to show that the derivative of a hypergeometric series can be expressed as follows:
$ \frac{d}{dx} {}\_nF\_m (a\_1,\ldots, a\_n;b\_1\ldots b\_m; x) = \frac {a\_1\cdots a\_n}{b\_1\cdots b\_m} {}\_nF\_m (a\_1+1,\ldots, a\_n+1;b\_1+1\ldots b\_m+1; x)$. From the other hand, for an arbitrary function $G(x)$ we have $(\log G(x))' = \frac {G'(x)}{G(x)}$.


It follows that it suffices to find a ratio of two hypergeometric series to find a logarithmic derivative. In some cases this ratio is known to be a hypergeometric series again. So after the integration we'll obtain the desired result.


Thank you in advance for any help.



Logarithm of a hypergeometric series
No. This is because all hypergeometrics are holonomic, and holonomic functions can only have a finite number of singularities, which themselves can only be of certain types. If the logarithm of all hypergeometrics could be so expressed, then you could have a holonomic function with a $\ln \ln (x)$ singularity, which is not possible.


I find the paper [On the non-holonomic character of logarithms, powers, and the nth prime function](http://algo.inria.fr/flajolet/Publications/FlGeSa05.pdf)  by Flajolet, Gerhold and Salvy (*The Electronic Journal of Combinatorics*, 2005, vol. 11) to be a wonderful compendium of useful tools for disproving holonomicity. Searching through the literature to find these tools is tedious, and so these authors ought to be commended for assembling so many into one pleasant paper.


