A conjectured integer sequence arising from the exponential integral

Richard P. Brent; M. L. Glasser; Anthony J. Guttmann

A conjectured integer sequence arising from the exponential integral

Richard P. Brent, M. L. Glasser, Anthony J. Guttmann

Research output: Contribution to journal › Article › peer-review

Abstract

Let f₀ (z) = exp(z/(1 − z)), f₁ (z) = exp(1/(1 − z))E₁ (1/(1 − z)), where (Formula Presented). Let a_n = [zⁿ]f₀ (z) and b_n = [zⁿ]f₁ (z) be the corresponding Maclaurin series coefficients. We show that a_n and b_n may be expressed in terms of confluent hypergeometric functions. We consider the asymptotic behaviour of the sequences (a_n) and (b_n) as n → ∞, showing that they are closely related, and proving a conjecture of Bruno Salvy regarding (b_n). Let (Formula Presented) is a Hadamard product. We obtain an asymptotic expansion 2n^3/2 ρ_n ∼ −∑ d_k n^−k as n → ∞, where d_k (Formula Presented) Q, d₀ = 1. We conjecture that 2^6k d_k (Formula Presented) Z. This has been verified for k ≤ 1000.

Original language	English
Article number	19.4.7
Journal	Journal of Integer Sequences
Volume	22
Issue number	4
Publication status	Published - 2019

Cite this

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title = "A conjectured integer sequence arising from the exponential integral",

abstract = "Let f0 (z) = exp(z/(1 − z)), f1 (z) = exp(1/(1 − z))E1 (1/(1 − z)), where (Formula Presented). 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 (Formula Presented) is a Hadamard product. We obtain an asymptotic expansion 2n3/2 ρn ∼ −∑ dk n−k as n → ∞, where dk (Formula Presented) Q, d0 = 1. We conjecture that 26k dk (Formula Presented) Z. This has been verified for k ≤ 1000.",

keywords = "Asymptotics, Confluent hypergeometric function, D-finite, Euler-Gompertz constant, Exponential integral, Hadamard product, Holonomic, Kummer function, Modified Bessel function, Stretched exponential",

author = "Brent, {Richard P.} and Glasser, {M. L.} and Guttmann, {Anthony J.}",

year = "2019",

language = "English",

volume = "22",

journal = "Journal of Integer Sequences",

issn = "1530-7638",

publisher = "University of Waterloo",

number = "4",

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TY - JOUR

T1 - A conjectured integer sequence arising from the exponential integral

AU - Brent, Richard P.

AU - Glasser, M. L.

AU - Guttmann, Anthony J.

PY - 2019

Y1 - 2019

N2 - Let f0 (z) = exp(z/(1 − z)), f1 (z) = exp(1/(1 − z))E1 (1/(1 − z)), where (Formula Presented). 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 (Formula Presented) is a Hadamard product. We obtain an asymptotic expansion 2n3/2 ρn ∼ −∑ dk n−k as n → ∞, where dk (Formula Presented) Q, d0 = 1. We conjecture that 26k dk (Formula Presented) Z. This has been verified for k ≤ 1000.

AB - Let f0 (z) = exp(z/(1 − z)), f1 (z) = exp(1/(1 − z))E1 (1/(1 − z)), where (Formula Presented). 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 (Formula Presented) is a Hadamard product. We obtain an asymptotic expansion 2n3/2 ρn ∼ −∑ dk n−k as n → ∞, where dk (Formula Presented) Q, d0 = 1. We conjecture that 26k dk (Formula Presented) Z. This has been verified for k ≤ 1000.

KW - Asymptotics

KW - Confluent hypergeometric function

KW - D-finite

KW - Euler-Gompertz constant

KW - Exponential integral

KW - Hadamard product

KW - Holonomic

KW - Kummer function

KW - Modified Bessel function

KW - Stretched exponential

UR - http://www.scopus.com/inward/record.url?scp=85079526922&partnerID=8YFLogxK

M3 - Article

SN - 1530-7638

VL - 22

JO - Journal of Integer Sequences

JF - Journal of Integer Sequences

IS - 4

M1 - 19.4.7

ER -

A conjectured integer sequence arising from the exponential integral

Abstract

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