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 f α ( x ) = e α - ( 1 + 1 / x ) α x , 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 ) = ∫ 0 ∞ e - s x φ α ( s ) d s , 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 ∞ and to obtain a very precise approximation of the largest α > 0 such that φ α ( s ) ≥ 0 , s > 0 , or equivalently, such that f α is completely monotonic.