a very occasional diary.




Zero is overrated

Define a sequence of functions \(P_i:\mathbb{R}^+\rightarrow\mathbb{R}\), \(i\in\mathbb{N}\)
$$P_0(x) = \ln(x)$$
$$P_{n+1}(x) = \int P_{n}(x)\cdot dx$$

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 \(P_i\):
$$P_1(x) = x\cdot\ln x - x$$
$$P_2(x) = \frac{x^2}{2}\cdot\ln x - \frac{3}{4}\cdot x^2$$
$$P_3(x) = \frac{x^6}{6}\cdot\ln x - \frac{11}{36}\cdot x^3$$

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

from where
$$K_n$$$$=$$$$\frac{1}{n}\cdot\Bigg(K_{n-1} + \frac{1}{n!}\Bigg)$$
$$=$$$$\frac{1}{n}\cdot\Bigg(\frac{1}{n!} + \frac{1}{n-1}\cdot\big(K_{n-2} + \frac{1}{(n-1)!}\big)\Bigg)$$
$$=$$$$\frac{1}{n!}\cdot\Bigg(\frac{1}{n} + \frac{1}{n-1}\Bigg) + \frac{1}{n\cdot(n - 1)}\cdot K_{n-2}$$
$$=$$$$\frac{1}{n!}\cdot\Bigg(\frac{1}{n} + \frac{1}{n - 1} + \frac{1}{n - 2}\Bigg) + \frac{1}{n\cdot(n-1)\cdot(n - 2)}\cdot K_{n-3}$$
$$=$$$$\frac{1}{n!}\cdot\Bigg(\frac{1}{n} + \frac{1}{n - 1} + \cdots +\frac{1}{n - p + 1}\Bigg) + \frac{1}{n\cdot(n-1)\cdot\cdots\cdot(n - p +1)}\cdot K_{n-p}$$
Substitute \(p = n - 1\):
$$K_n = \frac{1}{n!}\cdot\Bigg(1 + \frac{1}{2} + \cdots + \frac{1}{n}\Bigg)$$
Substitute this in the hypothesis:
$$P_n = \frac{x^n}{n!}\cdot\Bigg(\ln x - \big(1 + \frac{1}{2} + \cdots + \frac{1}{n}\big)\Bigg)$$
This nicely contains fragments of exponent, nth-harmonic number and, after a diagonalisation, the Euler constant:
$$\lim_{n\to +\infty}\frac{n!}{n^n}\cdot P_n(n) = -\gamma$$

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 \(m\):
$$(\forall n\in\mathbb{Z})(\forall m\in\mathbb{N})\partial^m P_n = P_{n - m}$$
and similarly
$$(\forall n\in\mathbb{Z})(\forall m\in\mathbb{N})\int_m P_n\cdot dx = P_{n + m}$$
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 \((\phi_i)_{i\in\mathbb{Z}}\) are there such that
$$(\forall m\in\mathbb{N})\partial^m\phi_n = \phi_{n - m}$$
$$(\forall m\in\mathbb{N})\int_m \phi_n\cdot dx = \phi_{n + m}$$
$$(\forall n\neq m)\phi_n \neq const\cdot\phi_m$$
(the latter condition is to avoid degenerate cases)?

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