

Preprint 87/2007
Ramanujan Eisenstein Series, Faá di Bruno Polynomials and Integrable Systems
Partha Guha and Dieter Mayer
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Submission date: 13. Sep. 2007
Pages: 17
Bibtex
MSC-Numbers: 58C20, 11C08
Keywords and phrases: Eisenstein series, Ramanujan differential equation, riccati equation, Faá di Bruno polynomial
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Abstract:
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.