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

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 φ_α, α> 0 , of entire functions such that fα(x)=∫0∞e-sxφα(s)ds,x>0. We show that each function φ_α has an expansion in power series, whose coefficients are determined in terms of Bell polynomials. This expansion leads to several properties of the functions φ_α, 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 φ_α 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 f_α is completely monotonic.