A Conjectured Integer Sequence Arising From the Exponential Integral
Brent, Richard; Glasser, M. L.; Guttmann, Anthony J
Description
Let f0(z)=exp(z/(1−z)), f1(z)=exp(1/(1−z))E1(1/(1−z)), where E1(x)=∫∞xe−tt−1dt. Let an=[zn]f0(z) and bn=[zn]f1(z) be the corresponding Maclaurin series coefficients. We show that an and bn may be expressed in terms of confluent hypergeometric functions. We consider the asymptotic behaviour of the sequences (an) and (bn) as n→∞, showing that they are closely related, and proving a conjecture of Bruno Salvy regarding (bn). Let ρn=anbn, so ∑ρnzn=(f0⊙f1)(z) is a Hadamard product. We obtain an...[Show more]
dc.contributor.author | Brent, Richard | |
---|---|---|
dc.contributor.author | Glasser, M. L. | |
dc.contributor.author | Guttmann, Anthony J | |
dc.date.accessioned | 2022-11-30T22:29:48Z | |
dc.identifier.issn | 1530-7638 | |
dc.identifier.uri | http://hdl.handle.net/1885/281430 | |
dc.description.abstract | Let f0(z)=exp(z/(1−z)), f1(z)=exp(1/(1−z))E1(1/(1−z)), where E1(x)=∫∞xe−tt−1dt. Let an=[zn]f0(z) and bn=[zn]f1(z) be the corresponding Maclaurin series coefficients. We show that an and bn may be expressed in terms of confluent hypergeometric functions. We consider the asymptotic behaviour of the sequences (an) and (bn) as n→∞, showing that they are closely related, and proving a conjecture of Bruno Salvy regarding (bn). Let ρn=anbn, so ∑ρnzn=(f0⊙f1)(z) is a Hadamard product. We obtain an asymptotic expansion 2n3/2ρn∼−∑dkn−k as n→∞, where the dk∈Q, d0=1. We conjecture that 26kdk∈Z. This has been verified for k≤1000. | |
dc.format.mimetype | application/pdf | |
dc.language.iso | en_AU | |
dc.publisher | University of Waterloo | |
dc.rights | © Journal of Integer Sequences | |
dc.source | Journal of Integer Sequences | |
dc.title | A Conjectured Integer Sequence Arising From the Exponential Integral | |
dc.type | Journal article | |
local.description.notes | Imported from ARIES | |
local.identifier.citationvolume | 22 | |
dc.date.issued | 2019 | |
local.identifier.absfor | 490409 - Ordinary differential equations, difference equations and dynamical systems | |
local.identifier.absfor | 490411 - Real and complex functions (incl. several variables) | |
local.identifier.absfor | 490303 - Numerical solution of differential and integral equations | |
local.identifier.ariespublication | u3102795xPUB5011 | |
local.publisher.url | https://cs.uwaterloo.ca/ | |
local.type.status | Published Version | |
local.contributor.affiliation | Brent, Richard, College of Science, ANU | |
local.contributor.affiliation | Glasser, M. L., Clarkson University | |
local.contributor.affiliation | Guttmann , Anthony J, University of Melbourne | |
local.description.embargo | 2099-12-31 | |
local.bibliographicCitation.issue | 4 | |
local.bibliographicCitation.startpage | 1 | |
local.bibliographicCitation.lastpage | 18 | |
dc.date.updated | 2021-11-28T07:30:13Z | |
local.identifier.thomsonID | WOS:000483311800001 | |
dcterms.accessRights | Free Access via publisher website | |
Collections | ANU Research Publications |
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