Zero is overrated

Define a sequence of functions $P_i:{\mathbb{R}}^{+}\to \mathbb{R}$, $i\in \mathbb{N}$

$$P_0(x)=\ln (x)$$ $$P_{n+1}(x)=\int P_n(x)\cdot dx$$

I found a beautiful closed formula for $P_i$, that I haven't seen before.

Integrating by parts, it's easy to calculate first few $P_i$:

$$\begin{array}{rcl}{P}_1(x)& =& x\cdot \ln x-x\\ P_2(x)& =& \frac{x^2}{2}\cdot \ln x-\frac{3}{4}\cdot x^2\\ P_3(x)& =& \frac{x^6}{6}\cdot \ln x-\frac{11}{36}\cdot x^3\end{array}$$

which leads to a hypothesis, that

$$P_n=\frac{x^n}{n!}\cdot \ln x-K_n\cdot x^n$$

for certain constants $K_i$, $K_1=1$.

Again integrating by parts, obtain:

$$K_{n+1}=\frac{1}{n+1}\cdot (K_n+\frac{1}{(n+1)!})$$

from where

$$\begin{array}{rcl}{K}_n& =& \frac{1}{n}\cdot (K_{n-1}+\frac{1}{n!})\\ & =& \frac{1}{n}\cdot (\frac{1}{n!}+\frac{1}{n-1}\cdot (K_{n-2}+\frac{1}{(n-1)!}))\\ & =& \frac{1}{n!}\cdot (\frac{1}{n}+\frac{1}{n-1})+\frac{1}{n\cdot (n-1)}\cdot K_{n-2}\\ & =& \frac{1}{n!}\cdot (\frac{1}{n}+\frac{1}{n-1}+\frac{1}{n-2})+\frac{1}{n\cdot (n-1)\cdot (n-2)}\cdot K_{n-3}\\ & =& \cdots \\ & =& \frac{1}{n!}\cdot (\frac{1}{n}+\frac{1}{n-1}+\cdots +\frac{1}{n-p+1})+\frac{1}{n\cdot (n-1)\cdot \cdots \cdot (n-p+1)}\cdot K_{n-p}\end{array}$$

Substitute $p=n-1$:

$$K_n=\frac{1}{n!}\cdot (1+\frac{1}{2}+\cdots +\frac{1}{n})$$

Substitute this in the hypothesis:

$$P_n=\frac{x^n}{n!}\cdot (\ln x-(1+\frac{1}{2}+\cdots +\frac{1}{n}))$$

This nicely contains fragments of exponent, nth-harmonic number and, after a diagonalisation, the Euler constant:

$$\underset{n\to +\mathrm{\infty }}{lim}\frac{n!}{n^n}\cdot P_n(n)=-\gamma$$

Why $P_i$ are interesting at all? Because if one completes them for negative indices as

$$P_n=(-1{)}^{n-1}\cdot (-n-1)!\cdot x^n$$

then mth-derivative of $P_n$ is $P_{n-m}$ for all non-negative $m$:

$$(\mathrm{\forall }n\in \mathbb{Z})(\mathrm{\forall }m\in \mathbb{N}){\partial }^m{P}_n=P_{n-m}$$

and similarly

$$(\mathrm{\forall }n\in \mathbb{Z})(\mathrm{\forall }m\in \mathbb{N}){\int }_m{P}_n\cdot dx=P_{n+m}$$

where ${\int }_m$ is repeated integral.

This is in contrast with powers $x^n$, $n\ge 0$, which, under repeated derivation, eventually pathetically collapse to a constant and then to 0, so that negative powers are not reachable from positive and other way around.

It's interesting, which other families $({\varphi }_i{)}_{i\in \mathbb{Z}}$ are there such that

$$(\mathrm{\forall }m\in \mathbb{N}){\partial }^m{\varphi }_n={\varphi }_{n-m}$$ $$(\mathrm{\forall }m\in \mathbb{N}){\int }_m{\varphi }_n\cdot dx={\varphi }_{n+m}$$

and

$$(\mathrm{\forall }n\ne m){\varphi }_n\ne const\cdot {\varphi }_m$$

(the latter condition is to avoid degenerate cases)?