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Consider the differential equation:

$$xf'(x) = f(x-1)$$

for $f : \mathbb{R} \to \mathbb{R}$. This has a solution given by $(1+x)$. However, this solution does not work for me, as I require a (non-zero) solution which satisfies $\lim_{x \to \infty} f(x) = 0$. This is due to the fact that $f$ here represents the derivative of an ansatz for a probability density function.

Which are the other solutions to this differential equation, and/or how can I solve to find them?

Edit: I now remembered the existence of the Fourier transform. If we apply this to both sides, we get:

$$i \frac{d}{dk} \hat{f'}(k) = e^{-ik} \hat{f}(k)$$

$$- \frac{d}{dk} \bigg( k \hat{f}(k) \bigg)= e^{-ik} \hat{f}(k)$$

$$\implies \hat{f} + k \frac{d \hat{f}}{dk} = -e^{-ik} \hat{f}$$

$$\implies \frac{d \hat{f}}{dk} = \frac{-(1 + e^{-ik})}{k} \hat{f}$$

I recall that the Fourier transform is a bijection if $f, \hat{f}$ are integrable. Of course, $1+x$ is not integrable.

Gonçalo
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J. S.
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    What you are asking about is very close to the Dickman function, https://en.wikipedia.org/wiki/Dickman_function which is the solution to the differential-delay equation $xf'(x)+f(x-1)=0$, $f(x)=1$ for $0\le x\le1$. – Gerry Myerson Jun 16 '24 at 03:06
  • @GerryMyerson Thank you for the reference, which I suppose means this differential equation will likewise not have a closed-form solution. I found this differential equation when considering uniform random variables, so it is a little surprising it looks so similar to a differential equation with number-theoretic significance. It doesn't seem to say in the article; do you know if this function is smooth? – J. S. Jun 16 '24 at 03:12
  • @J.S. Yes, the function is smooth. You can tell because $f' (x) = \frac{f (x - 1)}{x}$ is the ratio of two continuous functions (where we exclude $x = 0$) and is therefore continuous. In a similar fashion, higher-order derivatives of $f$ can be expressed as a ratio of (sums of terms of the form $x^n f (x - k)$, where $n, k \in \mathbb{Z}$). – K. Jiang Jun 16 '24 at 04:47
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    @K.Jiang Of course; I should have seen that myself. Thank you! – J. S. Jun 16 '24 at 14:59
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    For those interested, I was deriving parts of Section 4 in this paper, which I found after Gerry's comment. I made a sign error; the differential equation is actually the one for the Dickman function, and it describes the probability density for which I was looking. – J. S. Jun 16 '24 at 15:03

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