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

## TL;DR

This paper constructs a family of entire functions linking the Bessel and Lambert W functions to analyze the complete monotonicity of a parameterized exponential function, using Fourier and complex analysis.

## Contribution

It introduces a new family of entire functions related to Bessel and Lambert W functions, providing a novel integral representation and properties relevant to complete monotonicity.

## Key findings

- Derived power series expansion for the functions lpha_\u03b1
- Identified the relationship between lphand well-known special functions
- Estimated the maximum lphaor which _lphare completely monotonic

## Abstract

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.