While doing some caculation related to von Neumann entropy, I encountered this kind of convergent series.
$$\text{Exl}(x) \equiv \sum _{n=1}^{\infty } \frac{x^n \log (n!)}{n!}$$
In my calculation, this function Exl$(x)$ appears in some places where exponential function should be, for example,
$$\frac{\cosh (x) \text{Cxl}(x) + \sinh (x)\text{Sxl}(x)}{\cosh(2x)}$$
appears in my calculation, where
$$\text{Cxl}(x) \equiv \frac{\text{Exl} (x) +\text{Exl} (-x)}{2}$$
and
$$\text{Sxl}(x) \equiv \frac{\text{Exl} (x) -\text{Exl} (-x)}{2}$$
are defined from the similarity to the hyperbolic functions.
As the given function Exl$(x)$ looks like some kind of 'augmented' exponential function as the following plot suggests,
I suspect there's a well defined special function related to this series. Is it so? Any kind of suggestion is appreciated.
Not really. Basically, $\text{Exl}(x)=-F'(1),$ where $F(k)=\displaystyle\sum_{n\ge0}\frac{x^n}{n!^k}~.~$ The only known values
of F are $F(0)=\dfrac1{1-x},~F(1)=e^x,$ and $F(2)=I_0\big(2\sqrt x\big).$ See Bessel function for more
information. Neither the function F, let alone its derivative, F', have ever been studied for
general values of the argument k. Alternately, we can use Stirling's approximation, but
neither $\displaystyle\sum_{n\ge1}\frac{x^n}{n!}\cdot\ln n$ nor $\displaystyle\sum_{n\ge0}\frac{x^n}{n!}\cdot n^k$ are expressible in terms of any known functions, be
they special or elementary. Well, that's not exactly true; the latter series yields the values
of Bell numbers for $x=1:$ see Dobinski's formula for more information.