Examples of closed forms of integrals with a power tower argument using W-Lambert function.

Question

Here is a closed form of an integral that looks like:

Integral form(s) of a general tetration/power tower integral solution: $$\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(an+b,cn+d)}{Γ(An+B)}$$

Using this identity and the series expansion of the Lambert W function and digamma function:

$$\text W(x)\mathop=^{|x|<\frac1e}\sum_{n=1}^\infty \frac{(-n)^{n-1}x^n}{n!}=-\int_1^\infty\lfloor x\rfloor d\left( \frac{(-n)^{n-1}x^n}{n!}\right)dn=-\int_0^\infty\lfloor n\rfloor \frac{(-n)^{n-1}x^n\left(1-\frac1n +\ln(-n)+\ln(x)-ψ(n+1)\right)}{n!}dn= \int_0^\infty \frac{\lfloor n\rfloor (-n)^{n-1}x^n }{nn!}dn-(\ln(x)+1)\int_0^\infty \frac{\lfloor n\rfloor (-n)^{n-1}x^n }{n!}dn -\int_0^\infty \frac{\lfloor n\rfloor\ln(-n) (-n)^{n-1}x^n }{n!}dn+\int_0^\infty \frac{\lfloor n\rfloor ψ(n+1) (-n)^{n-1}x^n }{n!}dn $$

Here is a graph of the integrand:

With this example when $x=\frac 13$. What are some other integrals that can be derived with the W-Lambert function with a power tower expression in the integral? One idea is to use the Abel-Plana formula on $\sum\limits_{n=1}^\infty \frac{(-n)^{n-1}x^n}{n!}$ if it works. Please correct me and give me feedback!

Answer 1

I know this question is a bit repetitive, but I could not ignore writing the following result using the Abel-Plana formula because it really works. Here is the final result:

$$\text W(x)\mathop=^{|x|<\frac 1e}\sum_{n=0}^\infty \frac{(-1)^n (n+1)^n x^{n+1}}{n!}=\frac t2+ \frac {it}2\int_0^\infty \frac{(it+1)^{it}x^{it}\text{csch}(\pi x)}{(i t+1)!}dt-\frac{i t}2\int_0^\infty \frac{(1-it)^{-it}x^{-it}\text{csch}(\pi x)}{(1-i t)!}dt $$

Now try to derive some other results using this one with substitutions etc. Please correct me and give me feedback!