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

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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.

Original languageEnglish
JournalConstructive Approximation
Issue number4
Pages (from-to)121–154
Publication statusPublished - 2021

    Research areas

  • Bell polynomials, Completely monotonic function, Complex analysis, Fourier analysis, Stieltjes moment sequence

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