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


Authors: Ahmad El-Guindy and Aleksandar Petrov
Journal: Proc. Amer. Math. Soc. 143 (2015), 3303-3318
MSC (2010): Primary 11F52, 11G09, 11M38
DOI: https://doi.org/10.1090/proc/12406
Published electronically: April 28, 2015
MathSciNet review: 3348773
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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; and a new proof of relations between special values of Pellarin $ L$-series.


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Additional Information

Ahmad El-Guindy
Affiliation: Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
Address at time of publication: Texas A&M University at Qatar, Science Program, Doha 23874, Qatar
Email: a.elguindy@gmail.com

Aleksandar Petrov
Affiliation: Texas A&M University at Qatar, Science Program, Doha 23874, Qatar
Address at time of publication: Max Planck Institute for Mathematics, vivatsgasse 7, 53111 Bonn, Germany
Email: apetrov@mpim-bonn.mpg.de

DOI: https://doi.org/10.1090/proc/12406
Keywords: Vectorial Drinfeld modular forms, $\tau$-recurrent sequences, deformations of Eisenstein series, $A$-expansions
Received by editor(s): May 12, 2013
Received by editor(s) in revised form: October 13, 2013
Published electronically: April 28, 2015
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society