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 \(\alpha>0\) for which \(f_\alpha(x)=e^\alpha - (1+1/x)^{\alpha x},\ x>0\) is a completely monotonic function, we combine Fourier analysis with complex analysis to find a family \(\varphi_\alpha\), \(\alpha>0\), of entire functions such that \(f_\alpha(x) =\int_0^\infty e^{-sx}\varphi_\alpha(s)\,ds, \ x>0.\) We show that each function \(\varphi_\alpha\) has an expansion in power series, whose coefficients are determined in terms of Bell polynomials. This expansion leads to several properties of the functions \(\varphi_\alpha\), 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 \(\varphi_\alpha\) as \(\alpha\) increases from \(0\) to \(\infty\) and to obtain a very precise approximation of the largest \(\alpha>0\) such that \(\varphi_\alpha(s)\geq0,\, s>0\), or equivalently, such that \(f_\alpha\) is completely monotonic.