What is the integral $\int x^t/\Gamma(1+t) \, dt$? (In general: relation between series and integrals)

Question

(The question arises from playing with translating series into integrals)

I wanted to see, what it means to have a "continuous" relative for powerseries and other series; the most simple one perhaps $$ \begin{array} {} f_1(x) = \sum _{k=0}^\infty x^k = {1 \over 1-x} &\to & g_1(x)= \int_0^\infty x^t \,dt = - { 1\over \log(x) } \end{array}$$ I couldn't get this $$ \begin{array} {} f_2(x)=\sum _{k=0}^\infty {x^k \over k!} = \exp(x) &\to& g_2(x)=\int_0^\infty { x^t\over \Gamma(1+t) } \,dt =\text{ ???} \end{array}$$ by, for instance, Wolfram alpha...
and I'd like to proceed to some more general $$ \begin{array} {} f_\varphi(x)=\sum _{k=0}^\infty \varphi(k) x^k &\to& g_\varphi(x)=\int_0^\infty \varphi(t) x^t \,dt =\text{ ???} \end{array}$$ where $\varphi(k)$ is some meaningful function producing common sets of coefficients for the power series.

Playing a bit with numerical evaluations for $f_2(x),g_2(x)$ so far did not uncover anything obvious, but I am interested, whether there are some relations known; for instance whether there are relations between $g_2(x) \cdot g_2(y)$ perhaps analoguously to $ f_2(x) \cdot f_2(y) = f_2(x+y)$ or the like.

Is there something known about it? Is there a possibly a list of sums/integrals-relations done elsewhere?

In general, I'd like to get more intuition about this; it reminds me that I should possibly re-read in the explanations for the Euler/MacLaurin-summation formula where the "dance between discrete and continuous" (as it is a title of a nice article about the work of Delabaere on Euler's divergent series) has a similar relevance. (But this is possibly too much for this Q&A-site ...)

Answer 1

Here's a thought: rewrite $g_2$ as

$$\int_0^{\infty} dt \frac{e^{-t \log{(1/x)}}}{t \Gamma(t)}$$

Now, $\frac1{t \Gamma(t)}$ is an entire function and therefore has the following Taylor series representation in the whole complex plane:

$$\frac1{t \Gamma(t)} = \sum_{k=0}^{\infty} a_k t^k$$

where $a_0=1$, $a_1=\gamma$, and for $k \ge 2$, the $a_k$ satisfy

$$a_k = \frac{a_1 a_{k-1}-\sum_{j=2}^{k} (-1)^j \zeta(j) a_{k-j}}{k} $$

Integrate and get the following

$$g_2(x) = \sum_{k=0}^{\infty} a_k \frac{k!}{\left ( \log{\frac1{x}}\right )^{k+1}} $$

Answer 2

One more relation by W/A: $$ \begin{array} {} f_{1,1}(x) = \sum _{k=1}^\infty k \cdot x^{k-1} = {1 \over (1-x)^2} &\to & g_{1,1}(x)= \int_1^\infty t \cdot x^{t-1} \,dt = { 1 - \log(x) \over \log^2(x) } \\& & \text{for Re} (\log(x)) \lt 1 \end{array}$$