Euler‘s factorial series at algebraic integer points 

Author:  Seppälä, Louna^{1} 
Organizations: 
^{1}Research Unit of Mathematical Sciences, University of Oulu, P.O. Box 8000, 90014, Finland 
Format:  article 
Version:  accepted version 
Access:  open 
Online Access:  PDF Full Text (PDF, 0.3 MB) 
Persistent link:  http://urn.fi/urn:nbn:fife202003309642 
Language:  English 
Published: 
Elsevier,
2020

Publish Date:  20200330 
Description: 
AbstractWe study a linear form in the values of Euler’s series \(F(t)=\sum\nolimits_{n=0}^\infty n!t^n\) at algebraic integer points \(α_j∈\mathbb{Z}_\mathbb{K}, j=1,…,m\), belonging to a number field \(\mathbb{K}\). In the two main results it is shown that there exists a nonArchimedean valuation \(v\vert p\) of the field \(\mathbb{K}\) such that the linear form \({\mathrm\Lambda}_v=\lambda_0+\lambda_1F_v(\alpha_1)+\dots+\lambda_mF_v(\alpha_m)\), \(\lambda_i\in{\mathbb{Z}}_\mathbb{K}\), does not vanish. The second result contains a lower bound for the vadic absolute value of \({\mathrm\Lambda}_v\), and the first one is also extended to the case of primes in residue classes. On the way to the main results, we present explicit Padé approximations to the generalised factorial series \(\sum\nolimits_{n=0}^\infty{\left(\prod\nolimits_{k=0}^{n1}P(k)\right)}t^n\), where \(P(x)\) is a polynomial of degree one. see all

Series: 
Journal of number theory 
ISSN:  0022314X 
ISSNE:  10961658 
ISSNL:  0022314X 
Volume:  206 
Pages:  250  281 
DOI:  10.1016/j.jnt.2019.06.013 
OADOI:  https://oadoi.org/10.1016/j.jnt.2019.06.013 
Type of Publication: 
A1 Journal article – refereed 
Field of Science: 
111 Mathematics 
Subjects:  
Funding: 
The work of the author was supported by the University of Oulu Scholarship Foundation and the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters. 
Copyright information: 
© 2019 Elsevier Inc. This manuscript version is made available under the CCBYNCND 4.0 license http://creativecommons.org/licenses/byncnd/4.0/. 
https://creativecommons.org/licenses/byncnd/4.0/ 