Dear Manuel and Doron,

I read with interest your article on the arXiv on Onsager's solution to the Ising model. Please allow me to give a little personal history on this problem.

In 1971 when I was a post-doc at King's College, University of London, Martin Sykes mentioned to me that computationally, the most precise and rapid way to
obtain the coefficients of the spontaneous magnetisation of the Ising model was from the linear recurrence relation satisfied by the coefficients. We laughed and
said that if we had the coefficients in the 1940s that we had now (in the 1970s) we could have found the solution experimentally (or even 20 years earlier, based on the coefficients known at that time). We speculated whether this was perhaps also true for the free-energy?

Together with my room-mate, Geoff Joyce, we considered this question, and derived the linear recurrence relation from Onsager's solution. This made me realise that searching for such recurrences (equivalent to fitting to a D-finite ODE) represented a powerful new method of series analysis.

We wrote this up in 1972 "On a new method of series analysis in lattice statistics", J.Phys A:Gen. Phys. vol. 5, L81, 1972 (G.S. Joyce and A. J. Guttmann).

Application of the method to a number of test series, and the explicit connection with Onsager's solution was then discussed in my paper "On the recurrence relation method of series analysis", J.Phys A:Gen. Phys. vol. 8, 1081, 1975 (A. J. Guttmann).

Both Joyce and I gave seminars on this at the time, and pointed out that we could have conjectured Onsager's solution based on our known series coefficients at that time.

Of course this is not the same as what you have done, where you've made clever use of symmetries, and choice of a natural variable, but it's philosophically along the same lines, and I thought you might be interested to know what was done on this topic some 46 years ago.

Best wishes,

Tony