By coincidence, half an hour ago I was reading Beukers, Frits. Hypergeometric functions, how special are they?Notices Amer. Math. Soc. 61 (2014), no. 1, 48--56. MR3137256 which discusses solving this quintic. I don't think it answers your question, but it is a very nice paper! There are further references in http://oeis.org/A002294 Neil On Wed, Mar 12, 2014 at 4:42 AM, Ahmad El-Guindy <a.elguindy@gmail.com>wrote:
Stillwell's paper `Eisenstein's Footnote' (link below) gives a lucid explanation of how Eisenstein (at the tender age of 14) obtained a solution y for y^5+y=x as an infinite series in x of the form
y=x-x^5+(10*x^9)/2!-(14*15*x^13)/3!+....
Furthermore, the Bring-Jerrard Theorem asserts that a general quintic can be reduced to one of the above form, with x a radical expression of the coefficient of said quintic.
http://web.a.ebscohost.com/ehost/pdfviewer/pdfviewer?sid=9dd1fb0a-49ef-42b5-...
Nonetheless, I don't think this gives the whole picture since the range of convergence for x isn't that big. Namely, the above series is exactly x*4F3(1/5, 2/5, 3/5, 4/5; 5/4, 2/4, 3/4|5^5*x^4/4^4), which converges (absolutely) for |x| less than or equal to (0.8)*(0.2)^(1/4), which is approximately 0.53499.
So, if one is interested in solving y^5+y=1, say, then the series above won't converge. Is there an analytic continuation of the 4F3 that would be defined for large values of x and which would still give a solution of that quintic, or is there another way to get around that?
Cheers,
Ahmad _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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