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

Christian Berg, Eugenio Massa, Ana P. Peron

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…