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Ramanujan Eisenstein Series, Faá di Bruno Polynomials and Integrable Systems
Partha Guha and Dieter Mayer
At first we express the higher order Riccati equation or Fa´a di Bruno polynomial in terms of the modified Ramanujan differential equations in analogy to the relation of the Chazy III equation and the well known Ramanujan equations for the Eisenstein series of the modular group. We relate Ramanujan’s series connected with the pentagonal numbers, introduced by Ramanujan in his Lost Notebook, to the Fa´a di Bruno polynomials and the Riccati chain determined by the Eisenstein series of weight two for the modular group. As a first step to get an explicit expression for the general term in Ramanujan’s polynomial of degree k we derive a formula for the n-th order differential equations this Eisenstein series fulfill.