[seqfan] Ramanujan's Nomes q1, q2 & q3.
Brad Klee
bradklee at gmail.com
Wed Jul 25 04:57:14 CEST 2018
Heya Seqfans,
On P. 46 of "Modular Equations and Approximations to Pi",
Ramanujan defines nomes q1, q2, q3 in an analogy with
the Jacobi nome q0 ( cf. http://oeis.org/A005797 ).
Let us work in Berndt's signature convention where q(2),
q(3), q(4), q(6) correspond to Ramanujan's q0, q2, q1, q3,
and variable "x" will be the G.f. expansion parameter.
We can check with A005797 that
q(2)/(x*(1-x)*2F1(1/2,1/2,1,x)^2) - d/dx[ q(2) ] = 0.
According to my most rigorous geometric calculations,
the following eigenvalue equation applies for all s=2,3,4,6 :
q(s)/(x*(1-x)*2F1(1/s,(s-1)/s,1,x)^2) - d/dx[ q(s) ] = 0.
We can readily solve this equation in series expansion.
========q(2), 16^n========
0, 1, 8, 84, 992, 12514, 164688 . . .
========q(3), 27^n========
0, 1, 15, 279, 5729, 124554 . . .
========q(4), 64^n========
0, 1, 40, 1876, 95072, 5045474 . . .
========q(6), 432^n========
0, 1, 312, 107604, 39073568 . . .
As far as I know, OEIS does not have series expansions
for Ramanujan's q1, q2, q3. If these calculations are
correct it seems they should be entered. If they are
in fact wrong, perhaps someone can find the correct
integers and enter those ASAP!
Cheers,
Brad
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