Re: [math-fun] q^20? Convolution and continued fractions
On Fri, Dec 9, 2011 at 5:04 AM, Bill Gosper <billgosper@gmail.com> wrote:
Mike Hirschhorn points out that since a regular CF with geometrically progressing terms can be normalized to have geometrically progressing numerators and constant denominators, e.g.,
1 c + ----------------- c 1 - + ------------- q c 1 -- + -------- 2 c 1 q -- + --- 3 . q . . = q (--) 2 c c (1 + ------------) 3 q (--) 2 c 1 + -------- 5 q (--) 2 c 1 + ---- . . . his paper, web.maths.unsw.edu.au/~mikeh/webpapers/paper12.pdf<http://web.maths.unsw.edu.au/%7Emikeh/webpapers/paper12.pdf>gives alternative expressions for the geometric denominators CF below (which include the Rogers-Ramanujan CF).
In which case my sum quotient lhs developed a rash of q^(1/4), leading to this peculiarity: Sum[q^n^2/QPochhammer[q, q, 2*n], {n, 0, Infinity}]/ Sum[q^(2*n + n^2)/QPochhammer[q, q, 1 + 2*n], {n, 0, Infinity}] == (QPochhammer[q^8, q^20]* QPochhammer[q^12, q^20])/(QPochhammer[q^4, q^20]* QPochhammer[q^16, q^20]) q^20? --rwg (Much aided by Julian) The general
formula in the mitslides.pdf allows denominator(k) = a/q^k + b, which seems neither fish nor fowl. --rwg
On Thu, Dec 8, 2011 at 1:43 PM, Bill Gosper <billgosper@gmail.com> wrote:
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:
[bagbiting wrap-happy GMail.]
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.) [...]
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 Sat, Dec 10, 2011 at 9:52 PM, Bill Gosper <billgosper@gmail.com> wrote:
On Fri, Dec 9, 2011 at 5:04 AM, Bill Gosper <billgosper@gmail.com> wrote:
Mike Hirschhorn points out that since a regular CF with geometrically progressing terms can be normalized to have geometrically progressing numerators and constant denominators, e.g.,
1 c + ----------------- c 1 - + ------------- q c 1 -- + -------- 2 c 1 q -- + --- 3 . q . . = q (--) 2 c c (1 + ------------) 3 q (--) 2 c 1 + -------- 5 q (--) 2 c 1 + ---- . . . his paper, web.maths.unsw.edu.au/~mikeh/webpapers/paper12.pdf<http://web.maths.unsw.edu.au/%7Emikeh/webpapers/paper12.pdf>gives alternative expressions for the geometric denominators CF below (which include the Rogers-Ramanujan CF).
In which case my sum quotient lhs developed a rash of q^(1/4), leading to this peculiarity:
Sum[q^n^2/QPochhammer[q, q, 2*n], {n, 0, Infinity}]/ Sum[q^(2*n + n^2)/QPochhammer[q, q, 1 + 2*n], {n, 0, Infinity}] == (QPochhammer[q^8, q^20]* QPochhammer[q^12, q^20])/(QPochhammer[q^4, q^20]* QPochhammer[q^16, q^20])
q^20? --rwg (Much aided by Julian) [...]
According to PSLQ, Sum[q^n^2/QPochhammer[q, q, 2*n],{n, 0, Infinity}]== 1/(QPochhammer[q, q^20]* QPochhammer[q^3, q^20]*QPochhammer[q^4, q^20]* QPochhammer[q^5, q^20]*QPochhammer[q^7, q^20]* QPochhammer[q^9, q^20]*QPochhammer[q^11, q^20]* QPochhammer[q^13, q^20]*QPochhammer[q^15, q^20]* QPochhammer[q^16, q^20]*QPochhammer[q^17, q^20]* QPochhammer[q^19, q^20]) and therefore Sum[q^(2*n + n^2)/QPochhammer[q, q, 1 + 2*n],{n, 0, Infinity}]== 1/(QPochhammer[q, q^20]* QPochhammer[q^3, q^20]*QPochhammer[q^5, q^20]* QPochhammer[q^7, q^20]*QPochhammer[q^8, q^20]* QPochhammer[q^9, q^20]*QPochhammer[q^11, q^20]* QPochhammer[q^12, q^20]*QPochhammer[q^13, q^20]* QPochhammer[q^15, q^20]*QPochhammer[q^17, q^20]* QPochhammer[q^19, q^20])} Are these known? (They could simplify slightly.) --rwg
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Bill Gosper