MikeS> Great! I'd love to see the rest of the slides. They're pretty cryptic: http://gosper.org/mitslides.pdf MikeS> I take it the denominator in each sum is the generalized q-Pochhammer symbol described at the bottom of the page here:http://mathworld.wolfram.com/q-PochhammerSymbol.html Precisely. (eqn 18) Here's the lhs written out: (c104) PRODCONTRACT(INTOPROD(MAKEPROD(D94[1]))); 2 n n - 1 2 -- - n/2 /===\ n n 2 | | 1 (- 1) %i q | | -------------------------------------- | | i i inf i = 0 %i (- 1) q i i + 1 ==== (------------ + 1) ((- 1) q + 1) \ c (c + %i) > ------------------------------------------------------------------ / n ==== c n = 0 (d104) --------------------------------------------------------------------------------- 2 n n - 1 2 -- + n/2 /===\ n 2 | | 1 %i q | | ------------------------------------------ | | i i + 1 inf i = 0 i i + 1 %i (- 1) q ==== ((- 1) q + 1) (1 - ----------------) \ c > --------------------------------------------------------------- / n ==== c n = 0 Numerically, for c=1, q=1/2, (c107) CF(RECTFORM(APPLY_NOUNS(POCHSIMP(APPLY_NOUNS(SUBST([Q = 5.0b-1,C = 1,INF = 15],D94[1])))))); (d107) [1, 2, 4, 8, 16, 32, 64, 128, 260] (This container was filled at the manufacturing facility, but some settling may have occurred during shipping.) An interesting partial fractions identity in those slides is Sum[(x + a)^-k/k*x^(k - n)/(n - k), {k, 1, n - 1}]== Sum[a^(k - n)/(n - k)/ k*((-1)^(n - k)/Binomial[n, k] - k/n)*(1/(x + a)^k + (-1)^(n - k)/x^k), {k, 1, n - 1}] but Mma turns it into a useless pile of garbage before I can feed it to << HolonomicFunctions.m (whose authors have since kindly shown me how to smuggle into their subfunctions before Mma finds out it has Sums.). On Thu, Dec 8, 2011 at 5:46 AM, Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote:> I just stumbled upon slides from some forgotten talk containing a matrix> sketch of> an identity which specializes to> 2> n> inf 2 -- - n/2> ==== n n 2> \ (- 1) %i q> (c + %i) > ---------------------> / n %i> ==== c (- --, - q; - q)> n = 0 c n> ------------------------------------> 2> n> inf 2 -- + n/2> ==== n 2> \ %i q> > --------------------> / n %i q> ==== c (- q, ----; - q)> n = 0 c n>> => 1> c + -----------------> c 1> - + -------------> q c 1> -- + --------> 2 c 1> q -- + ---> 3 .> q .> .>>> (Sorry, no Mma. It was being unusably, inconceivably stupid on this.)> --rwg>> On Thu, Dec 1, 2011 at 10:59 AM, Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote:>>> On Wed, Nov 30, 2011 at 2:51 PM, Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com> <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote:> This promised to be quite tedious, except that it was possible to take>> "unconscionable> shortcuts" (I think I called them). I think I can dig up some results, if you want.>>>>>> Yes, please; I don't get any hits on searching my archive for>> "unconscionable" except this email.>> -->> Mike Stay - metaweta@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=metaweta%40gmail.com> <http://gosper.org/webmail/src/compose.php?send_to=metaweta%40gmail.com>http://www.cs.auckland.ac.nz/~mikehttp://reperiendi.wordpress.com>>>> You can pretty much find everything searching for ContinuedFractionK. But>> I see a big garble in a msg containing>> [...] And constant/linear gives 1F0>>>>>> ContinuedFractionK[e, a n + b, {n, 1, Infinity}] ==>>>> b e>> e Hypergeometric0F1[2 + -, --]>> a 2>> a>> ------------------------------------,>> b e>> (a + b) Hypergeometric0F1[1 + -, --]>> a 2>> a>> and a pattern emerges. But not quite.[...]>> ------------->> The pattern in question predicted the values of p and q in the pFqs based>> on>> the degrees of the two polynomials in the ContinuedFractionK, to which I'd>> found a>> strange exception. Unfortunately, I seem to have spazzed the editing and>> stated>> the same example twice in succession.>> --rwg>> I think the aforementioned shortcuts somehow combined the determination of>> the>> ODE integration constants with the subsequent limit-taking.>>