TY - JOUR

T1 - A Family of Entire Functions Connecting the Bessel Function J1 and the Lambert W Function

AU - Berg, Christian

AU - Massa, Eugenio

AU - Peron, Ana P.

PY - 2021

Y1 - 2021

N2 - Motivated by the problem of determining the values of α> 0 for which fα(x)=eα-(1+1/x)αx,x>0, is a completely monotonic function, we combine Fourier analysis with complex analysis to find a family φα, α> 0 , of entire functions such that fα(x)=∫0∞e-sxφα(s)ds,x>0. We show that each function φα has an expansion in power series, whose coefficients are determined in terms of Bell polynomials. This expansion leads to several properties of the functions φα, which turn out to be related to the well-known Bessel function J1 and the Lambert W function. On the other hand, by numerically evaluating the series expansion, we are able to show the behavior of φα as α increases from 0 to ∞ and to obtain a very precise approximation of the largest α> 0 such that φα(s)≥0,s>0, or equivalently, such that fα is completely monotonic.

AB - Motivated by the problem of determining the values of α> 0 for which fα(x)=eα-(1+1/x)αx,x>0, is a completely monotonic function, we combine Fourier analysis with complex analysis to find a family φα, α> 0 , of entire functions such that fα(x)=∫0∞e-sxφα(s)ds,x>0. We show that each function φα has an expansion in power series, whose coefficients are determined in terms of Bell polynomials. This expansion leads to several properties of the functions φα, which turn out to be related to the well-known Bessel function J1 and the Lambert W function. On the other hand, by numerically evaluating the series expansion, we are able to show the behavior of φα as α increases from 0 to ∞ and to obtain a very precise approximation of the largest α> 0 such that φα(s)≥0,s>0, or equivalently, such that fα is completely monotonic.

KW - Bell polynomials

KW - Completely monotonic function

KW - Complex analysis

KW - Fourier analysis

KW - Stieltjes moment sequence

UR - http://www.scopus.com/inward/record.url?scp=85078230670&partnerID=8YFLogxK

U2 - 10.1007/s00365-020-09499-x

DO - 10.1007/s00365-020-09499-x

M3 - Journal article

AN - SCOPUS:85078230670

VL - 53

SP - 121

EP - 154

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

IS - 4

ER -