Exponential integral $ \int_0^\infty \frac{x^t}{\Gamma(t+1)}\text dt$

Question

Now since the sum $$ \sum_{n=0}^\infty \frac{x^n}{n!},\quad x\in\Bbb R, $$ does have some relatively nice properties, is the same true for its analogues integral? If we take the gamma function to be a generalisation of the factorial with $\Gamma(n+1) = n!$, an obvious analogues integral formula would be $$ \int_0^\infty \frac{x^t}{\Gamma(t+1)}\text dt,\quad x\ge 0. $$ Does this integral have any similar, nice properties?

Answer 1

The integral can be rephrased, using the transformation $x=e^x$ in the same spirit as the general analogy of Taylor series to Laplace transforms, as $$ \int_0^\infty \frac{x^t \mathrm dt}{\Gamma(t+1)} =\int_0^\infty \frac{e^{st}\mathrm dt}{\Gamma(t+1)}. $$ This integral is known as the nu function, denoted $\nu(x)=\nu(e^s)$. Wikipedia and MathWorld have the same definition, but don't offer much detail. For more information, the best place to go is probably Erdelyi et al., Higher Trascendental Functions vol. 3, p.217, §18.3 (where chapter 18 is just 'miscellaneous functions'). Quoting from there:

[The function $\nu(x)$] was encountered by Volterra in his theory of convolution-logarithms (Volterra 1916 Chapter VI, Volterra and Péres 1924, Chapter X) [...]. These functions also occur in connection with operational calculus, appear in an inversion formula of the Laplace transformation, and are of interest in connection with certain integral equations.

Erdelyi et al. show that $$ \nu(x)=\begin{cases} e^x+O(|x|^{-N}) & |\arg(x)|\leq\pi/2 \\ O(|x|^{-N}) & \pi/2<|\arg(x)|\leq\pi \end{cases} $$ for any integer $N$, and that apart from being a simple(ish) Laplace transform itself, $\nu(x)$ has a simple Laplace transform, $$ \int_0^\infty e^{-st}\nu(t)\mathrm dt=\frac{1}{s\log(s)} \quad\operatorname{Re}(s)>1. $$ Both of these properties are (apparently) relevant for the use of $\nu(x)$ in operational calculus; Erdelyi et al. give references from the 1940s but presumably the field has moved on since then.

Regarding the relation to integral equations, the nu function obeys the rather pleasing equation $$ \int_0^\infty \exp\left(-\frac{x^2}{4y}\right)\nu(x)\mathrm dx =\frac{\sqrt{\pi y}}{2}\nu(y), $$ i.e. it is an eigenfunction of the integral kernel $y^{-1/2}\exp\left(-\frac{x^2}{4y}\right)$, solving an integral equation which "gives all characteristic functions which, in a certain sense, are of regular growth", though it is apparently a solution to other interesting equations of similar form.

For further details, see the references in Erdelyi et al.