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Here is a set of difference equations for y[n]

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

or equivalently

4 (3 + 2 n) (5168520 + n (42053906 + 5 n (28672669 + 5 n (10621126 + 5 n (2308917 + 10 n (147271 + 20 n (2551 + 370 n))))))) + (-8 (1 + n) (3 + 2 n) (8 + 5 n) (7 + 10 n) (9 + 10 n) (11 + 10 n) (13 + 10 n) (11 + 20 n)) y[n] + 3 (2 + 5 n) (3 + 5 n)^2 (4 + 5 n) (6 + 5 n) (5 + 6 n) (7 + 6 n) (31 + 20 n) y[1 + n] == 0, y[1] == 158/31

I would appreciate it if a reader could provide the Maple solution to these equations--as I have been analyzing it with Mathematica, and obtaining a "hypergeometric-rich" solution of large complexity.

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Maple doesn't return any hypergeometrics. Its solution has a lot of Gamma functions and a finite summation:

-(3+5*n)*Pi^(1/2)*GAMMA(n+13/10)*GAMMA(n+11/10)*GAMMA(n+9/10)*GAMMA(n+7/10)*GAMMA(n+3/2)*(-cos((1/5)*Pi)+2*cos((1/5)*Pi)^2-1)*64^n*27^(-n)*GAMMA(1+n)*(Sum((4/3)*(cos((1/5)*Pi)+1)*(2*cos((1/5)*Pi)-1)*GAMMA(n1+7/5)*GAMMA(n1+9/5)*GAMMA(n1+11/6)*GAMMA(n1+11/5)*GAMMA(n1+13/6)*GAMMA(n1+8/5)*(2*n1+3)*(9250000*n1^7+63775000*n1^6+184088750*n1^5+288614625*n1^4+265528150*n1^3+143363345*n1^2+42053906*n1+5168520)/(GAMMA(2+n1)*27^(-n1-1)*64^(n1+1)*(2*cos((1/5)*Pi)+1)*(cos((1/5)*Pi)-1)*GAMMA(n1+5/2)*GAMMA(n1+17/10)*GAMMA(n1+19/10)*GAMMA(n1+21/10)*GAMMA(n1+23/10)*Pi^(1/2)*(8+5*n1)*(6*n1+5)*(5*n1+2)*(5*n1+4)*(5*n1+6)*(6*n1+7)*(5*n1+3)^2), n1 = 1 .. n-1)+7900/1287)/((20*n+11)*GAMMA(n+3/5)*GAMMA(n+7/6)*GAMMA(n+6/5)*GAMMA(n+5/6)*GAMMA(n+4/5)*GAMMA(n+2/5)*(cos((1/5)*Pi)+2*cos((1/5)*Pi)^2-1))
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  • $\begingroup$ Thanks very, very much Carl Love! I find that very interesting. Might I ask you to send me a copy of the output at slater@kitp.ucsb.edu? (And/or is it inappropriate/unsuitable to post the output itself on the stack exchange?) $\endgroup$ – Paul B. Slater Aug 26 '16 at 13:51
  • $\begingroup$ This equation pertains to issues in quantum-information and random matrix theory (arxiv.org/find/quant-ph/1/au:+Slater_P/0/1/0/all/0/1). It is the first (k=1) in a series of increasingly larger systems (for which I presently also have k=2,3). I strongly suspect that this Maple output simplifies greatly, in particular if it is multiplied by a factor (which I tried to post in LaTeX form--but led to too many characters in the comment). $\endgroup$ – Paul B. Slater Aug 26 '16 at 15:21
  • $\begingroup$ I emailed you a PDF transcript of the Maple session. I don't know what you mean by "post the output itself." Doesn't my Answer above contain "the output itself"? Please give my Answer a Vote Up. $\endgroup$ – Carl Love Aug 26 '16 at 15:56
  • $\begingroup$ OK, thanks again (got the email)! I hadn't (stupidly) seen how to observe and grab the complete output using the black bar. I pasted this into a Maple (14) worksheet and will try to simplify it there, and will try it on my other (presently) two (increasingly large) difference equations. I will also try to input it to Mathematica (my more "native" language). But maybe it's not entirely clear at this point that the summation in the formula will not also yield hypergeometric results (as I obtained in Mathematica). Because of length limits, I have an immediately next (website) comment too. $\endgroup$ – Paul B. Slater Aug 26 '16 at 16:23
  • $\begingroup$ This difference equation has also been the subject of an earlier (Mathematica) stack exchange item: mathematica.stackexchange.com/questions/124342/… $\endgroup$ – Paul B. Slater Aug 26 '16 at 16:25

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