Re: [math-fun] q^20? Convolution and continued fractions
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 More than slightly. Noodling just now with Julian, Sum[q^n^2/QPochhammer[q, q, 2*n],{n, 0, Infinity}]== QPochhammer[q^2, q^2]/ (QPochhammer[q, q]*QPochhammer[q^4, q^20]* QPochhammer[q^16, q^20]) Sum[q^(2*n + n^2)/QPochhammer[q, q, 1 + 2*n], {n, 0, Infinity}] == QPochhammer[q^2, q^2]/ (QPochhammer[q, q]*QPochhammer[q^8, q^20]* QPochhammer[q^12, q^20]) Still, no word from the toldyasos. --rwg
On Sat, Dec 17, 2011 at 4:54 PM, Bill Gosper <billgosper@gmail.com> wrote:
his paper, web.maths.unsw.edu.au/~mikeh/webpapers/paper12.pdf<http://web.maths.unsw.edu.au/%7Emikeh/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
More than slightly. Noodling just now with Julian, Sum[q^n^2/QPochhammer[q, q, 2*n],{n, 0, Infinity}]== QPochhammer[q^2, q^2]/ (QPochhammer[q, q]*QPochhammer[q^4, q^20]* QPochhammer[q^16, q^20])
Sum[q^(2*n + n^2)/QPochhammer[q, q, 1 + 2*n], {n, 0, Infinity}] == QPochhammer[q^2, q^2]/ (QPochhammer[q, q]*QPochhammer[q^8, q^20]* QPochhammer[q^12, q^20])
Still, no word from the toldyasos. --rwg
Oops, Mike "kindly" points out that these are (2.20.1) and (2.20.2) in ftp://ftp2.de.freebsd.org/pub/EMIS/journals/EJC/Surveys/ds15.pdf --rwg Man, there are pochhammers base q^7 in there.
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Bill Gosper