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MiS Preprint

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.

Sep 13, 2007
Sep 13, 2007
MSC Codes:
58C20, 11C08
Eisenstein series, Ramanujan differential equation, riccati equation, Faá di Bruno polynomial

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2007 Repository Open Access
Partha Guha and Dieter Mayer

Ramanujan Eisenstein series, Faá di Bruno polynomials and integrable systems