What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$?

Question

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

Answer 1

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.