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
- 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.
- 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.
- 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.
- 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.
- D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946.
- 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.
- https://math.stackexchange.com/a/4262516/945479
- https://math.stackexchange.com/a/4262498/945479
Best Answer
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