Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.
MiS Preprint
87/2007
Ramanujan Eisenstein Series, Faá di Bruno Polynomials and Integrable Systems
Partha Guha and Dieter Mayer
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.