MaplePrimes - comments on Post, A hypergeom challenge

MaplePrimes - comments on Post, A hypergeom challenge http://www.mapleprimes.com/posts/38790-A-Hypergeom-Challenge en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Tue, 09 Jun 2026 18:29:18 GMT Tue, 09 Jun 2026 18:29:18 GMT The latest comments added to the Post, A hypergeom challenge http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Post, A hypergeom challenge http://www.mapleprimes.com/posts/38790-A-Hypergeom-Challenge Very tough http://www.mapleprimes.com/posts/38790-A-Hypergeom-Challenge?ref=Feed:MaplePrimes:A hypergeom challenge:Comments#comment70007 My attempts to crack this have not worked, but perhaps some of my ideas might be useful to someone else. My favourite tactic is to transform such problems into DE problems, so <pre> K := (81/539)*42^(1/2)*hypergeom([-1/2, 3/8, 169/88], [81/88, 19/8], -x/6): de := collect(numer((lhs-rhs)(op([1,1],PDEtools[dpolyform](y(x)=K,no_Fn)))),diff,factor);</pre> <maple>-507*y(x)+(3334*x+18468)*diff(y(x),x)+32*x*(211*x+1134)*diff(diff(y(x),x),x)+1408*x^2*(6+x)*diff(diff(diff(y(x),x),x),x)</maple> 3rd order, as expected (but the stupid display of the <maple> tag is annoying here). The coefficients are not too nasty, and nicely the sqrt(42) has magically disappeared. Taking a look at the solution: <pre> ans := rhs(dsolve(de));</pre> <maple>_C1/x^(11/8)*(-1+x)+_C2*Int((6+x)^(1/2)*(-81+25*x)*x^(3/8)/(-1+x)^2,x)*(-1+x)/x^(11/8)+_C3*Int(Int(1/x^(81/88)*(-1+x)/(-81+25*x)^2/(6+x)^(3/2),x)*(6+x)^(1/2)*(-81+25*x)*x^(3/8)/(-1+x)^2,x)*(-1+x)/x^(11/8) </maple> [Again, the output in Classic is much nicer than here]  Where does this come from? Well, the DE, as an operator, factors: <pre> > DEtools[DFactor](DEtools[de2diffop](de,y(x),[Dx,x]),[Dx,x]); 2 2 3 48 (2875 x + 5911 x - 27378) x [(8448 x + 1408 x ) Dx + -------------------------------, -81 + 25 x 2 3 -1863 x + 82 x + 25 x + 972 -11 + 3 x Dx + ---------------------------------, Dx + ------------] 2 x (-1 + x) (6 + x) (-81 + 25 x) 8 x (-1 + x) </pre> Trying to figure out the values of the constants _C1, _C2 and _C3 turns out to be very hard since all 3 terms are singular at 0, so one would have to find out how to 'match' those singularities, which I have not been able to do. However, it might be useful to know that the problem is "reduced" to one involving algebraic integrals. I have not tried to use any of the contiguity relations or quadratic transformations formulas for the 3F2 to see if somehow that can cause some of the arguments to 'collapse' and thereby simplify the problem. It's too bad that Maple does not have a 'transformations' package with these formulas as transformations one can apply. The latest comments added to the Post, A hypergeom challenge 70007 Mon, 25 Aug 2008 18:00:12 Z JacquesC JacquesC thought similar http://www.mapleprimes.com/posts/38790-A-Hypergeom-Challenge?ref=Feed:MaplePrimes:A hypergeom challenge:Comments#comment82939 and run into the 3rd oder DE (without splitting), it would not help to provide initial conditions here, but have not tried in points different from 0 BTW the task seems to stem from the problems seen in <a href="http://www.mapleprimes.com/forum/thomascalculus5615exercisegetshypergeom">www.mapleprimes.com/forum/thomascalculus5615exercisegetshypergeom</a> where it was unclear how to simplify the 2F1 The latest comments added to the Post, A hypergeom challenge 82939 Mon, 25 Aug 2008 18:22:49 Z Axel Vogt Axel Vogt oh http://www.mapleprimes.com/posts/38790-A-Hypergeom-Challenge?ref=Feed:MaplePrimes:A hypergeom challenge:Comments#comment93985 When seeing Edgar's challenge problem, my first thought was convert(W,Int) and to try working from there. The latest comments added to the Post, A hypergeom challenge 93985 Mon, 25 Aug 2008 18:26:56 Z pagan pagan has not worked for me http://www.mapleprimes.com/posts/38790-A-Hypergeom-Challenge?ref=Feed:MaplePrimes:A hypergeom challenge:Comments#comment93986 has not worked for me, got just a double integral, it should stand for 3F2 =Int(2F1) = Int(Int(...)) using Euler's integral I guess ... and similar for trying sums :-( The latest comments added to the Post, A hypergeom challenge 93986 Mon, 25 Aug 2008 18:30:15 Z Axel Vogt Axel Vogt guess http://www.mapleprimes.com/posts/38790-A-Hypergeom-Challenge?ref=Feed:MaplePrimes:A hypergeom challenge:Comments#comment93987 My guess was that, somehow, the inner integrand might be the simplified result of adding two terms (perhaps each of which would then work in `int`, while the sum might not). I guessed this on the weak grounds that I thought that I saw hints in Edgar's posts in the other thread that you cited. The latest comments added to the Post, A hypergeom challenge 93987 Mon, 25 Aug 2008 18:35:19 Z pagan pagan "Cheating?" w/ Maple http://www.mapleprimes.com/posts/38790-A-Hypergeom-Challenge?ref=Feed:MaplePrimes:A hypergeom challenge:Comments#comment70000 The challenge was to show that W:=1.  There was no mention as to what available tools one could use to do this.  The methods outlined above are over my head mathematically.   Attached file showing W:=1. <a href="http://www.mapleprimes.com/files/221_hypergeom.mws">Download 221_hypergeom.mws</a> <a href="http://www.mapleprimes.com/viewfile/2752">View file details</a>   Regards, Georgios Kokovidis <address>Dräger Medical</address> <pre> </pre> The latest comments added to the Post, A hypergeom challenge 70000 Mon, 25 Aug 2008 19:38:04 Z gkokovidis gkokovidis