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

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 ($1\le l\le q$) 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.

arXiv comment16 pages