An interesting recurrent equality, possibly easier to solve in its differential form?

Question

I encountered an interesting inequality that I'm not sure how to approach. Here $c$ is a positive constant. $$f(n+1) - f(n) = c f(n)\sum_{m=0}^n f(m)$$

I am not familiar with techniques to solve recursive equations, so I thought about a similar differential equation of it $$\frac{\partial f}{\partial x} = cf(x)\int_{z=0}^x f(z)dz.$$

I am quite new to the topic of differential equations though, so I might be missing some obvious techniques to solve this kind of equality. With my limited knowledge I tried applying Laplace transform or the Fourier transform to this equation to no avail, but the expression looks nice enough that I suspect there is an analytical solution to it. Any help to getting an analytical expression of this equation is much appreciated!

(This is coming from a real-world problem, so for now, any "nice" assumption on $f$ and the initial conditions and so on can be placed)

Answer 1

Your question comes in two variants. One is discrete. The other continuous.

The discrete variant asks about the equation $$ f(n+1) - f(n) = c f(n)\sum_{m=0}^n f(m). \tag{1} $$ A nicer version of this is to define $$ a_n := c f(n), \qquad b_n := 1 + \sum_{m=0}^n a_m. \tag{2} $$ Multiply equation $(1)$ by $\,c\,$ to get $$ a_{n+1} = a_n + a_n\sum_{m=0}^n a_m = a_n b_n = a_0\prod_{m=0}^n b_m \tag{3} $$ where the initial value $\,a_0\,$ is arbitrary. A simple example is if $\,a_0=1.\,$ This is OEIS sequence A001697. The growth rate is given as $\,a_n \sim c^{2^n}\,$ where $\,c \approx 1.335245.\,$ This is the typical growth rate with a different constant for each choice of $\,a_0.\,$ I don't think there is a known closed form.

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The continuous variant asks about the related differential equation $$ \frac{\partial f}{\partial x} = cf(x)\int_{z=0}^x f(z)dz. \tag{4} $$ This is more interesting. The equation is equivalent to $$ \frac{\partial g}{\partial x} = c\int_{z=0}^x e^{g(z)}dz \;\; \text{ where } \;\; g(x) := \log(f(x)). $$ Assuming $\,g(x)\,$ has a power series, let $$ g(x) = a_0 + a_2\frac{x^2}{2!} + a_4\frac{x^4}{4!} + \cdots $$ where $\,c = a_2/e^{a_0}.\,$ For simplicity assume that $\,a_0=0, a_2=1.\,$ The solution of equation $(4)$ is $$ f(x) = \sec\left(\frac{x}{\sqrt{2}}\right)^2 = 1\frac{x^2}{2!} +4\frac{x^4}{4!} +34\frac{x^6}{6!} + \cdots $$ where the coefficients are OEIS sequence A002105.

Answer 2

Being a combination of product and sum, it looks difficult that there might be a closed form.

The best I can suggest is to make the substitution

$$ f(n) = 2^{g(n)} $$ I am using $2$ as a base because $2^n$ has a simple difference.

Then $$ \eqalign{ & f(n + 1) - f(n) = 2^{g(n + 1)} - 2^{g(n)} = 2^{g(n)} \left( {2^{g(n + 1) - g(n)} - 1} \right) \cr & 2^{g(n)} \left( {2^{g(n + 1) - g(n)} - 1} \right) = c2^{g(n)} \sum\limits_{k = 0}^n {2^{g(k)} } \cr & 2^{g(n + 1) - g(n)} = 1 + c\sum\limits_{k = 0}^n {2^{g(k)} } \cr & 2^{g(n + 2) - g(n + 1)} - 2^{g(n + 1) - g(n)} = c2^{g(n + 1)} \cr & 2^{g(n + 2) - 2g(n + 1)} - 2^{ - g(n)} = c \cr & \left( {2^{g(n + 2) - 2g(n + 1) + g(n)} - 1} \right)2^{ - g(n)} = c \cr & 2^{\Delta ^2 g(n)} = 1 + c2^{g(n)} \cr & 2^{\Delta ^2 g(n)} = 1 + 2^{\ln c + g(n)} \cr & \approx \Delta ^2 g(n) + g(n) = \ln c\quad \left| {\;1 < < 2^{g(n)} } \right. \cr} $$

So the asymptotics is clear.