I wonder what work have been done on the zeros of the digamma function and on the values of the gamma function at such points (on the negative real axis). Any help please :)

1$\begingroup$ I've fixed the tags and added a link. I think you might like to consult the FAQs on how to ask a really good question. In particular, you might provide some background on where you are coming from and what you know already. $\endgroup$– David Roberts ♦Commented Aug 3, 2012 at 0:41
1 Answer
See, for example, P. Sebah, X. Gourdon, Introduction to the Gamma Function, available here.
Topic 5.1.5, page 13, is about Zeros of the digamma function. We can see that on the negative axis, the digamma function has a single zero between each consecutive negative integers (the poles of the gamma function).
The authors presents the first five zeros of the digamma function on the negative axis with 50 decimal places.
ADDED:
I have found this beautiful manuscript written by Hermite, with reference to gamma functions, Cours de M. Hermite, Librairie Scientifique A. Hermann, 1883.
Also, the complete Oeuvres de Charles Hermite is available here.
Another reference is NIST Digital Library of Mathematical Functions

$\begingroup$ In a research paper that I did I found something which is equivalent to what is shown on pg 13 of that article: x_n ~ n + 1/logn. And if we let d_n denote the absolute value of the gamma function at those points, then I found that (n!d_n)/logn = e, when n goes to infinity. Now my professor wants me to relate these results to the mathematical world by citing the similar results that others have found before me. I saw that the author of that article says that the first result was mentioned by Hermite but do you by any chance know the specific article or book of Hermite that has that? Thanks!!! $\endgroup$– Tri NgoCommented Aug 3, 2012 at 14:47

$\begingroup$ @Tri Ngo: Please, see ADDED in the above answer. $\endgroup$– PapiroCommented Aug 3, 2012 at 16:48

$\begingroup$ Thanks PaPiro, I wonder if they have ever translated his work into English. $\endgroup$– Tri NgoCommented Aug 6, 2012 at 14:18