functional equation $ 1= f(x)+\left(\frac x2\right)+\dots+f\left(\frac xN\right) $

Given the equation $$ 1= f(x)+f\left(\frac x2\right)+f\left(\frac x3\right)+f\left(\frac x4\right) $$ how could I solve it, or the more general equation $$ 1= f(x)+\left(\frac x2\right)+f\left(\frac x3\right)+f\left(\frac x4\right)+\dots+f\left(\frac xN\right) $$ for a given number $N$? Could I use Wolfram math online to solve it?

for the case $$ 1= f(x)+f(x-2)+f(x-3)+f(x-4) $$ I could find a generating function since it is just a recurrence equation.

Solution 1:

If we assume that the function is analytic, then the first observation to make is that $$ \lim_{x\to0} \sum_{i=1}^N f(x/i)=N\lim_{x\to0}f(x) = 1\\ \lim_{x\to0}f(x) = \frac1N $$ Now, if you take the derivative, you have $$ \sum_{i=1}^N \frac1if'(x/i) = 0 $$ In the limit as $x\to0$, we have $$ \lim_{x\to0}\sum_{i=1}^N \frac1if'(x/i) = \sum_{i=1}^N \frac1i \cdot \lim_{x\to0}f'(x) = 0\\ \lim_{x\to0}f'(x) = 0 $$ And this process can be repeated for all derivatives. As such, as it's analytic, we find that $f(x)=\frac1N$ is the only analytic solution.

If we don't require analyticity, there may be many more solutions - at the very least, I expect some interesting everywhere-discontinuous functions.

Solution 2:

Let $r$ be an non-negative integer. Let $\lambda_k = \alpha_k + \beta_k i$ be any set of roots of the equation:

$$ 1 + 2^{-\lambda} + 3^{-\lambda} + 4^{-\lambda} + \cdots + N^{-\lambda} = 0\tag{*1}$$ subject to the constraint $\alpha_i > r$ and $\beta_i > 0$.

For any real numbers $A_k^{\pm}$ and $\delta_k^{\pm}$, consider the function $g$ defined by:

$$g(x) = \begin{cases} \sum_{k} A_k^{+} \Re\left(e^{i\delta_k^{+}} x^{\lambda_k} \right)\\ \sum_{k} A_k^{-} \Re\left(e^{i\delta_k^{-}} |x|^{\lambda_k} \right) \end{cases} = \begin{cases} \sum_{k} A_k^{+} x^{\alpha_k}\cos(\beta_k \log x + \delta_k^{+}), &\quad\text{ for } x \ge 0\\ \sum_{k} A_k^{-} |x|^{\alpha_k}\cos(\beta_k \log |x| + \delta_k^{-}),&\quad\text{ for } x \le 0 \end{cases} $$

It is easy to check $f(x) = g(x) + \frac{1}{N}$ satisfy the functional equation

$$1 = f(x) + f(\frac{x}{2}) + \cdots + f(\frac{x}{N})\tag{*2}$$

provided the sum over $k$ converges. Furthermore, if $A_k^{\pm}$ decrease fast enough, the constraint $\alpha_i > r$ will force $g$ and its first $r^{th}$ derivatives vanishes and continuous at $0$.

In short,

if the set of roots of $(*1)$ is non-empty, then there are non-trivial $C^{r}$ solution for $(*2)$.

For an example, consider the case $N = 4$ and $r = 0$. It seems there are infinite many solution for $(*1)$. The one with smallest $\beta$ is give by:

$$\lambda_1 = \alpha_1 + \beta_1 i \sim 0.62597108186373 + 3.127120203586539 i$$

Which leads to a non-trivial continuous solution for $(*2)$: $$f(x) = \frac{1}{4} + |x|^{\alpha_1} \cos(\beta_1 \log|x|) \sim \frac14 + |x|^{0.62597108186373} \cos( 3.127120203586539 \log |x|)$$ Please note that for $N = 4$, it seems all the complex roots of $(*1)$ has $\Re(\lambda) < 1$. It is not clear whether $(*2)$ has any non-trivial $C^{1}$ solution at all.

Solution 3:

Let $h(x)=f(e^x)-1/N$.

Then $$h(x)+h(x-\log 2) + h(x-\log 3) +\cdots + h(x-\log N)=0$$

This brings it close to the form of a recurrence relation, but is of a type I do not know how to even attempt to solve. I suspect that very little is actually known about such defining equations. (The only thing I could find was Recurrence relations on a continuous domain in which nothing was actually deduced in the end anyway!)

As FiveLemon points out, in fact one can seek $$h(x)=a^x$$ type solutions to this equation just as for recurrence relations. One finds a nonpolynomial equation in $a$ which is the natural generalization of the recurrence relation case. It's not immediately clear whether there are such solutions or whether these are in any sense whatsoever complete. (Yes? Probably not? resp.) The equation should be investigated! I don't have a computer atm...