On symmetric powers of $\tau$-recurrent sequences and deformations of Eisenstein series

TL;DR

This paper establishes key identities among $ au$-recurrent sequences linked to Drinfeld modular forms, leading to new insights into Eisenstein series deformations, $A$-expansions, and Pellarin $L$-series relations.

Contribution

It proves the equality of several $ au$-recurrent sequences and explores their implications for Eisenstein series deformations and Drinfeld modular forms.

Findings

An $A$-expansion for powers of deformed Eisenstein series.

Relations between Drinfeld modular forms with $A$-expansions.

A new proof of relations between Pellarin $L$-series values.

Abstract

We prove the equality of several $\tau$-recurrent sequences, which were first considered by Pellarin, and which have close connections to Drinfeld vectorial modular forms. Our result has several consequences: an $A$-expansion for the $l^\text{th}$ power ($1 \leq l \leq q$) of the deformation of the weight 2 Eisenstein series; relations between Drinfeld modular forms with $A$-expansions; a new proof of relations between special values of Pellarin $L$-series.