A Grammatical Calculus for the Ramanujan Polynomials

TL;DR

This paper introduces a grammatical calculus approach to Ramanujan polynomials, revealing their combinatorial structure and deriving key recurrence relations through a novel labeling scheme for rooted trees.

Contribution

It presents a new grammatical framework for Ramanujan polynomials, connecting combinatorial structures with recurrence relations using labeled rooted trees.

Findings

Developed a labeling scheme for rooted trees with improper edges.

Formulated a grammatical calculus for Ramanujan polynomials.

Derived recurrence relations from the grammatical framework.

Abstract

The Ramanujan polynomials arise in three intertwined contexts. As remarked by BerndtEvans-Wilson, no combinatorial perspective seems to be alluded to in the original definition of Ramanujan. On a different stage, Dumont-Ramamonjisoa uncovered a combinatorial structure underneath an equation also considered by Ramanujan. Around the same time, Shor came up with the same construction as a refinement of the classical formula of Cayley for trees. We present a labeling scheme for rooted trees by employing an extra label marking improper edges. Harnessed by this grammar, we develop a grammatical calculus for the Ramanujan polynomials heavily relying on the constant properties. Moreover, we provide a grammatical formulation of a correspondence that leads to the recurrence relation due to Berndt-Evans-Wilson and Shor.