[seqfan] Geometric G.F. for Ramanujan Periods
Brad Klee
bradklee at gmail.com
Thu Dec 21 16:48:25 CET 2017
Hi Seqfans,
After: http://list.seqfan.eu/pipermail/seqfan/2017-December/018157.html
The sequences A113424, A006480, A000897 we might call "Ramanujan Periods",
by an analogy with A002894. These are all hypergeometric series [1], and
periods in the sense of Kontsevich and Zagier (cf. Ref. [2], page 10 vs.
A897). It's worth mentioning that OEIS entries on Ramanujan Elliptic
Functions--and all publications I've seen--are incomplete in some regards.
The three Ramanujan periods are also period-energy functions in Hamiltonian
mechanics or symplectic geometry. The normal generating functions follow
from a geometry of plane curves, summarized by a "Geometric Generating
Function (G.G.F.)" ( cf. https://ptpb.pw/pR29 ) !
I realize that some people really do not want to draw any pictures;
however, the idea of G.G.F. provides a plausible middle-ground. It would be
nice to see a lot more of these recorded on OEIS, especially next to
integral and differential calculus. One difficulty is that the G.G.F. are
non-unique. The cubic-quartic shear symplectomorphism [3,4] between,
q^2 + p^2 - q^2*p <---> q^2 + p^2 - (1/4)*q^4
preserves area, thus provides two distinct means to generate the series for
A897 as the x-differential of area enclosed by surface contours f(q,p)=x.
This is not all, if A897(n) are coefficients to an even expansion in powers
x^(2n), we have two more, noticeably-distinct, quartic functions related by
a double-cover [5],
q^2 + p^2 - ( q^3*p - q*p^3 ) <---> (q^2 + p^2)*( 1 - (1/2)*q*p )
This is a similar situation as encountered with Edward's curve and A002894
[6,7].
But now it seems more the time to attack a few instances of walnut
geometry, so perhaps continued efforts on sequence and series can wait
until next year.
Happy Holidays,
Brad
[1] http://qseries.org/fgarvan/papers/alternativebases.pdf
[2] http://www.maths.ed.ac.uk/~aar/papers/kontzagi.pdf
[3] https://moderncrypto.org/mail-archive/curves/2017/000951.html
[4] https://www.youtube.com/watch?v=njZKDKQ_ttM
[5] https://www.youtube.com/watch?v=l2tRbvi--JQ
[6] https://moderncrypto.org/mail-archive/curves/2017/000950.html
[7] https://www.youtube.com/watch?v=7IqfauZ_9wQ
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