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