The proof of Holder's theorem for Gamma function(transcendentally transcendental) Ι am doing a project about the Gamma Function defined on the complex plane (on an undergraduate level), and I want to write the main properties of this function. One that I want to is that gamma function is transcendentally transcendental, which means it does not satisfy any algebraic differential equation. This is a theorem proved by Holder. Where can i find a proof which fits to my undergraduate level? I've searched on the internet, and I found some articles such as 'A Survey of Transcendentally Transcendental Functions' from The American Mathematical Monthly Vol. 96  or 'Concerning transcendentally transcendental functions'from Mathematische Annalen Vol. 48, but they are far away from my purpose. I think the proof in 'Ueber die Eigenschaft der Gammafunction keiner algebraischen Differentialgleichung zu genügen' 'Band 28 Mathematische Annalen' is the best for me, but I cannot understand it because I don't know German. Also, I found one on Wikipedia, but I don't think that is as strictly as I want. I would appreciate any help.
 A: I think this would be an excellent thesis precisely because there doesn’t seem to be any existing elementary expository account that includes a reasonably accessible detailed proof. In fact, if you can pull this off, I suspect your thesis would wind up being cited frequently on the internet in the same way as the case with Johan Thim’s Masters thesis Continuous Nowhere Differentiable Functions.
I did some digging, much more than is evident from what I found worth mentioning below, unfortunately, but for what it's worth here are a couple of references (which I imagine you probably already know about):
Eliakim Hastings Moore, Concerning transcendentally transcendental functions, Mathematische Annalen 48 (1897), 49-74.

This paper is in English and it is also less technical than later papers in English of which I am aware. I suspect someone at one time in the U.S. has written a Masters thesis that includes what you want to do, but I was not able to find one at the ProQuest archive (which is mainly Ph.D. dissertations, but some Masters theses are also included) or a similar treatment in someone's Ph.D. dissertation.

Philip J. Davis, Leonhard Euler’s integral: A historical profile of the gamma function, American Mathematical Monthly 66 #10 (December 1959), 849-869.

In the last complete paragraph on p. 864, the transcendentally transcendent nature of the gamma function is discussed, and mention is made that in 1925 the Russian mathematician Alexander Ostrowski gave an alternate proof of Hölder’s theorem. For the specific reference, and for references of some other alternate proofs (none in English besides Moore’s 1897 paper above, however), see the first paragraph of this paper.

