Aha! I just derived the Rogers-Fine identity as Sum[(t^k*QPochhammer[a, q, k])/QPochhammer[b, q, k], {k, 0, Infinity}] == Sum[(b^k*q^((-1 + k)*k)*t^k*(1 - a*q^(2*k)*t)*QPochhammer[a, q, k]* QPochhammer[(a*q*t)/b, q, k])/ (QPochhammer[b, q, k]*QPochhammer[t, q, 1 + k]), {k, 0, Infinity}] from a contour in the k,n plane, starting at i,j,k,n ={t,a,0,b}, with the unspecialized matrix system {{i, {{(q^(-1 + k + n)*(1 - q^(1 + i + j + k)))/(1 - q^(i + k)), (1 - q^(-1 + k + n))/(1 - q^(i + k))}, {0, 1}}}, {j, {{((1 - q^(1 + i + j + k))*(1 - q^(j + k + n)))/(1 - q^(1 + j)), -((q^(1 + j)*(1 - q^(-1 + k + n)))/(1 - q^(1 + j)))}, {0, 1}}}, {k, {{(q^(i + 2*k + n)*(1 - q^(1 + i + j + k))*(1 - q^(j + k + n)))/((1 - q^(i + k))*(1 - q^(k + n))), (1 - q^(i + j + 2*k + n))/(1 - q^(i + k))}, {0, 1}}}, {n, {{(q^(i + k)*(1 - q^(j + k + n)))/(1 - q^(k + n)), 1}, {0, 1}}}} I can generalize this out the wazoo, but mainly to matrix products which aren't sums. But I should have found this in AIM 304. --rwg n -> i + k j + k + n q (1 - q ) ----------------------- k + n 1 - q 1 0 1 k -> i + 2 k + n 1 + i + j + k j + k + n i + j + 2 k + n q (1 - q ) (1 - q ) 1 - q -------------------------------------------------- -------------------- i + k k + n i + k (1 - q ) (1 - q ) 1 - q 0 1 j -> 1 + i + j + k j + k + n 1 + j -1 + k + n (1 - q ) (1 - q ) q (1 - q ) ------------------------------------- -(------------------------) 1 + j 1 + j 1 - q 1 - q 0 1 i -> -1 + k + n 1 + i + j + k -1 + k + n q (1 - q ) 1 - q -------------------------------- --------------- i + k i + k 1 - q 1 - q 0 1 Although that product of Lucas numbers in my fib formula is just a q-pochhammmer, it provides theta-convergence because |q|>1. And since you'd sum both it and yours effectively via matrix product, it's more competitive than it looks. (E.g, you don't win using an addition chain on q^k^2 because you need (k-1)^2 etc. as well, so you just maintain q^(2k+1). * Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> [Feb 07. 2012 11:40]:> [jj:]> without splitting into even and odd part (yes, RWG, AIM304 exists).> > YOW, I'd forgotten that stunt! joerg>That one needed a small correction, see http://www.jjj.de/lambert-paper/ (Warning: very first iteration, just spend today over it; certainly contains typos (tell me if you spot one!)). What I referred to as "Fine's versatile relation"> Now this one turns out to be easily obtained by setting a := -b> in Fine's "versatile" relation (14-1) (p.15 in "Basic Hypergeometric> Series and Applications"). appears to be known as "Rogers-Fine identity", this term is used in my draft. Later I'll process your other mails (big THANKS btw.!)

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