A family of entire functions connecting the Bessel function $J_1$ and the Lambert $W$ function

Motivated by the problem of determining the values of $α>0$ for which $f_α(x)=e^α- (1+1/x)^{αx},\ x>0$ 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 $f_α(x) =\int_0^\infty e^{-sx}φ_α(s)\,ds, \ x>0.$ 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 $J_1$ 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 $\infty$ and to obtain a very precise approximation of the largest $α>0$ such that $φ_α(s)\geq0,\, s>0$, or equivalently, such that $f_α$ is completely monotonic.

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