A function $q(x)$ is said to be completely monotonic on an interval $I$ if $q(x)$ has derivatives of all orders on $I$ and $(-1)^{n}q^{(n)}(x)\ge0$ for $x\in I$ and $n\ge0$. See Chapter 1 in the monograph [1] below.


A positive function $q(x)$ is said to be logarithmically completely monotonic on an interval $I\subseteq\mathbb{R}$ if it has derivatives of all orders on $I$ and its logarithm $\ln q(x)$ satisfies $(-1)^k[\ln q(x)]^{(k)}\ge0$ for $k\in\mathbb{N}=\{1,2,\dotsc\}$ on $I$. See Definition 1 in th article [2] below.


A logarithmically completely monotonic function on $I$ must be completely monotonic on $I$, but not conversely. See Theorem 1 in [2] and related texts in the references [1, 3, 4] below.


The famous Bernstein-Widder's theorem (on page 161 Theorem 12b in the book [5]) reads that a necessary and sufficient condition that $q(x)$ should be completely monotonic for $0<x<\infty$ is that
\begin{equation} \label{berstein-1}\tag{w}
q(x)=\int\_0^\infty \textrm{e}^{-xt}\textrm{d}\,\alpha(t),
\end{equation}
where $\alpha(t)$ is non-decreasing and the integral \eqref{berstein-1} converges for $0<x<\infty$.


It is trivial that the exponential function $\textrm{e}^{1/x}$ is logarithmically completely monotonic on $(0,\infty)$. Consequently, by the above-mentioned Theorem 1 in [2], we conclude that the function $\textrm{e}^{1/x}$ is completely monotonic on $(0,\infty)$.


Motivated by the Bernstein-Widder's theorem mentioned above, we pose a question:


**What is the explicit expression of the measure $\alpha(t)$ such that
\begin{equation} \label{exp-frac1x}\tag{+}
\textrm{e}^{1/x}=\int\_0^\infty \textrm{e}^{-xt}\textrm{d}\,\alpha(t)
\end{equation}
converges for $0<x<\infty$?** See Section 4 in the paper [6] below.


References


1. R. L. Schilling, R. Song, and Z. Vondracek, *Bernstein Functions*, 2nd ed., de Gruyter Studies in Mathematics **37**, Walter de Gruyter, Berlin, Germany, 2012; available online at <https://doi.org/10.1515/9783110269338>.
2. F. Qi and C.-P. Chen, *A complete monotonicity property of the gamma function*, J. Math. Anal. Appl. **296** (2004), 603--607; available online at <https://doi.org/10.1016/j.jmaa.2004.04.026>.
3. C. Berg, *Integral representation of some functions related to the gamma function*, Mediterr. J. Math. **1** (2004), no. 4, 433--439; available online at <https://doi.org/10.1007/s00009-004-0022-6>.
4. B.-N. Guo and F. Qi, *A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function*, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. **72** (2010), no. 2, 21--30.
5. D. V. Widder, *The Laplace Transform*, Princeton University Press, Princeton, 1946.
6. Xiao-Jing Zhang, Feng Qi, and Wen-Hui Li, *Properties of three functions relating to the exponential function and the existence of partitions of unity*, International Journal of Open Problems in Computer Science and Mathematics **5** (2012), no. 3, 122--127; available online at <https://doi.org/10.12816/0006128>.
7. <https://math.stackexchange.com/a/4262516/945479>
8. <https://math.stackexchange.com/a/4262498/945479>



What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$?
For $k\in\mathbb{N}\_0=\{0,1,2,\dotsc\}$ and $z\ne0$, let
\begin{equation}\label{exp=k=sum-eq-degree=k+1}
H\_k(z)=\textrm{e}^{1/z}-\sum\_{m=0}^k\frac{1}{m!}\frac1{z^m}.
\end{equation}
For $\Re(z)>0$, the function $H\_k(z)$ has the integral representations
\begin{equation}\label{exp=k=degree=k+1-int}
H\_k(z)=\frac1{k!(k+1)!}\int\_0^\infty {}\_1F\_2(1;k+1,k+2;t)t^k \textrm{e}^{-zt}\textrm{d}\,t
\end{equation}
and
\begin{equation}\label{exp=k=degree=k+1-int-bes}
H\_k(z)=\frac1{z^{k+1}}\biggl[\frac1{(k+1)!}+\int\_0^\infty \frac{I\_{k+2} \bigl(2\sqrt{t}\,\bigr)}{t^{(k+2)/2}} \textrm{e}^{-zt}\textrm{d}\,t\biggr],
\end{equation}
where the hypergeometric series
\begin{equation}
{}\_pF\_q(a\_1,\dotsc,a\_p;b\_1,\dotsc,b\_q;x)=\sum\_{n=0}^\infty\frac{(a\_1)\_n\dotsm(a\_p)\_n} {(b\_1)\_n\dotsm(b\_q)\_n}\frac{x^n}{n!}
\end{equation}
for $b\_i\notin\{0,-1,-2,\dotsc\}$, the shifted factorial $(a)\_0=1$ and
\begin{equation}
(a)\_n=a(a+1)\dotsm(a+n-1)
\end{equation}
for $n>0$ and any real or complex number $a$, and the modified Bessel function of the first kind
\begin{equation}\label{I=nu(z)-eq}
I\_\nu(z)= \sum\_{k=0}^\infty\frac1{k!\Gamma(\nu+k+1)}\biggl(\frac{z}2\biggr)^{2k+\nu}
\end{equation}
for $\nu\in\mathbb{R}$ and $z\in\mathbb{C}$.


References


1. Feng Qi and Shu-Hong Wang, *Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions*, Global Journal of Mathematical Analysis **2** (2014), no. 3, 91--97; available online at <https://doi.org/10.14419/gjma.v2i3.2919>.
2. Feng Qi, *Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions*, Mathematical Inequalities & Applications **18** (2015), no. 2, 493--518; available online at <https://doi.org/10.7153/mia-18-37>.
3. Feng Qi and Christian Berg, *Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function*, Mediterranean Journal of Mathematics **10** (2013), no. 4, 1685--1696; available online at <https://doi.org/10.1007/s00009-013-0272-2>.
4. Bai-Ni Guo and Feng Qi, *Some integral representations and properties of Lah numbers*, Journal for Algebra and Number Theory Academia **4** (2014), no. 3, 77--87.
5. Xiao-Jing Zhang, Feng Qi, and Wen-Hui Li, *Properties of three functions relating to the exponential function and the existence of partitions of unity*, International Journal of Open Problems in Computer Science and Mathematics **5** (2012), no. 3, 122--127; available online at <https://doi.org/10.12816/0006128>.
6. Bai-Ni Guo and Feng Qi, *An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions*, Global Journal of Mathematical Analysis **2** (2014), no. 4, 243--248; available online at <https://doi.org/10.14419/gjma.v2i4.3310>.
7. Feng Qi and Xiao-Jing Zhang, *Complete monotonicity of a difference between the exponential and trigamma functions*, Journal of the Korea Society of Mathematical Education Series B: The Pure and Applied Mathematics **21** (2014), no. 2, 141--145; available online at <https://doi.org/10.7468/jksmeb.2014.21.2.141>.


