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Find RSolve solution reflecting special structure of DifferenceRoot[Function[{\[FormalY],...

Discussion in 'Mathematics' started by Paul B. Slater, Aug 1, 2020.

  1. 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?

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