# Find RSolve solution reflecting special structure of DifferenceRoot[Function[{\[FormalY], \[FormalN]}

RSolve yields a large (multi-page) solution (LeafCount=25891) containing a number of 7F6 (and higher) hypergeometric functions when applied to

DifferenceRoot[Function[{\[FormalY], \[FormalN]}, {62022240 + 545995032*\[FormalN] + 2056791388*\[FormalN]^2 +
4333244560*\[FormalN]^3 + 5587600700*\[FormalN]^4 + 4517982000*\[FormalN]^5 + 2238010000*\[FormalN]^6 +
621200000*\[FormalN]^7 + 74000000*\[FormalN]^8 + (-19027008 - 158454120*\[FormalN] - 566231672*\[FormalN]^2 -
1135130960*\[FormalN]^3 - 1397526400*\[FormalN]^4 - 1082880000*\[FormalN]^5 - 516080000*\[FormalN]^6 -
138400000*\[FormalN]^7 - 16000000*\[FormalN]^8)*\[FormalY][\[FormalN]] + 3*(2 + 5*\[FormalN])*(3 + 5*\[FormalN])^2*
(4 + 5*\[FormalN])*(6 + 5*\[FormalN])*(5 + 6*\[FormalN])*(7 + 6*\[FormalN])*(31 + 20*\[FormalN])*\[FormalY][1 + \[FormalN]] == 0,
\[FormalY][1] == 158/31}]][n]


But I think that RSolve is working here without respect to the apparent highly-structured form of the input (just essentially expanding everything out before further proceeding). But various subject-matter considerations lead me to conjecture that there is a much more compact/elegant solution. (Of course, one can try to simplify the actual very large solution--but previous experience indicates that is a most formidable--if even doable--task.)

So, my question here is how--if at all possible--one might incorporate the special structure of the DifferenceRoot expression in the RSolve (or some equivalent) process?

• Thanks J_Nat. Yes, that is precisely what I did--with the very large output (LeafCount=25891) being produced after (roughly) a few hours. – Paul B. Slater Aug 19 '16 at 19:08
• After about an hour FunctionExpand yields a result with a LeafCount of "only" 19429. It contains instances of HypergeometricPFQ with 54 different arguments. – bbgodfrey Aug 19 '16 at 19:37
• And, after an additional quarter-hour, Simplify[%, n \[Element] Integers && n > 0] reduces LeafCount to 13886. – bbgodfrey Aug 19 '16 at 20:11
• Thanks very much, bbgodfrey! I guess I could replicate your results, but maybe you could send them to me (slater@kitp.ucsb.edu). (I wasn't aware of the particular use of Simplify[]--interesting!) – Paul B. Slater Aug 19 '16 at 20:55
• Wow, I'm impressed and thankful for all your hard efforts, bbgodfey. I've been "messing around" for some time with big hypergeometric expressions strongly related, it would seem off-hand, to this one (see Fig. 3 in arxiv.org/pdf/1301.6617.pdf). Off-hand I'm not optimistic about the use of FullSimplify--but let's hope for something miraculous. – Paul B. Slater Aug 19 '16 at 21:59

As previously indicated, RSolve provides a large ("hypergeometric-function-abundant") solution (LeafCount=25981). Mathematica indicates that the solution naturally breaks into two parts--the first having a LeafCount of 5250. High-precision numerics, followed by rationalizations and use of the FindSequenceFunction command gives for this first part the relatively simple (hypergeometric-free) formula

(5*3^(-1 - 3*\[FormalN])*8^(1 + 2*\[FormalN])*(3 + 5*\[FormalN])*Pochhammer[7/10, \[FormalN]]*Pochhammer[9/10, \[FormalN]]*
Pochhammer[1, \[FormalN]]*Pochhammer[11/10, \[FormalN]]*Pochhammer[13/10, \[FormalN]]*
Pochhammer[3/2, \[FormalN]])/((11 + 20*\[FormalN])*Pochhammer[2/5, \[FormalN]]*Pochhammer[3/5, \[FormalN]]*
Pochhammer[4/5, \[FormalN]]*Pochhammer[5/6, \[FormalN]]*Pochhammer[7/6, \[FormalN]]*Pochhammer[6/5, \[FormalN]])
`

Now if we subtract its values from the original DifferenceRoot[Function, giving us the values of the second part, we, quite amazingly it would seem, get back exactly the original function but for the single change that y[1] now equals -4102/93, rather than 158/31 = 474/93. It would seem that this must be indicating something of a profound nature about the original function (and hopefully a relatively simple formula for it), but at present I don't see what that is.

• RSolve has now given me the solution with y[1]=-4102/93, instead of the original 158/31 = 474/93. Again, the solution naturally breaks into two parts, with exactly the same LeafCounts. But now, the first part (which had the relatively simple hypergeometric-free formula for it previously with the use of 158/31) turns out to be identically zero (as high-precision numerics convincingly indicate). So, the other part would simply yield the values used to produce the second DifferentialRoot[Function to which RSolve was just applied. – Paul B. Slater Aug 21 '16 at 16:33
• On the mathematics stack exchange site, I inquired as to how Maple might handle the equation--and was given a solution, presently "hypergeometric-free", but with a (yet unperformed) summation that conceivably might lead to hypergeometric functions. Here is the link math.stackexchange.com/questions/1903720/… – Paul B. Slater Aug 26 '16 at 16:29
• I did translate the Maple output of Love to Mathematica and performed the (implicit) summation. The result has a very similar nature to the one originally obtained in Mathematica, with the same set of hypergeometric terms, all with argument 27/64, appearing. So, it would seem that the two languages take similar approaches (as to which I was curious) to the recurrence equations. – Paul B. Slater Aug 26 '16 at 19:48