I found one formula connecting the logarithmic integral function, $li(x)$, to polynomials as following:
$$li(x) = \frac{x}{\ln(x)} + \sum_{k=1}^{+\infty} \frac{P_k(\ln(x))}{x^k}$$
where $P_k(x)$ is the $k^{th}$ Bell polynomial, which is a polynomial in $x$ that arises in the study of combinatorics and number theory.
This formula expresses $li(x)$ as a sum of terms involving powers of $\frac{1}{x}$ and polynomials in $\ln(x)$. The first term, $\frac{x}{\ln(x)}$, is the main term of the expansion, and it dominates the behavior of $li(x)$ as x becomes large. The remaining terms, which involve the Bell polynomials, represent "corrections" to the main term that become increasingly small as $x$ increases.
The Bell series expansion of $li(x)$ is a useful tool in the analysis of prime numbers.
My question is simple: if $P_k(x)$ refers to the $k^{th}$ Bell polynomial, then $x$ must be an integer. Nevertheless here we have ${P_k(\ln(x))}$. I'm looking for the conditions for $x$ when we define a function f derived from the formula above:
$$f_k(x)=\frac{x}{\ln(x)}+\sum_{k=1}^{+\infty} \frac{P_k(\ln(x))}{x^k} = li(x)$$
Thanks.