Two "incompatible" recurrence relations / functional equations: $f(y+10)=2\cdot f(y+3)-f(y)$ and $f(y+10)=\frac{10}9\cdot f(y+8)-\frac19\cdot f(y)$

Question

Imagine I suppose (for the sake of contradiction) a function $f(x)$ defined over $[0,\infty]$ for which three conditions hold:

1. $f(y+10)=2\cdot f(y+3)-f(y)$ for any $y\geq0$
2. $f(y+10)=\frac{10}{9}\cdot f(y+8)-\frac{1}{9}\cdot f(y)$ for any $y\geq0$
3. $f$ is strictly increasing

My intuition is that these statements are incompatible: there does not exist a $f$ that satisfies all conditions. My sense is that (from the answer to a previous question):

• (1) and (3) implies the function is of the form ($-\beta\cdot e^{-0.180107\cdot x}$ + cyclic components)
• (2) and (3) implies that the function is of the form ($-\beta\cdot e^{-0.150939\cdot x}$ + cyclic components)
• No function can satisfy both of these forms.

I can maybe prove this for these specific equations, but I really want to prove a more general statement that if we posit that:

1. $f$ satisfies functional equation / recurrence relation 1
2. $f$ satisfies functional equation / recurrence relation 2
3. $f$ is strictly increasing

and there are no overlapping real roots in the characteristic equation of relation 1 and 2, there is a contradiction.

However, I am not sure exactly how to prove this. Any help would be appreciated. Very happy to hear intutions, references, or a proof sketch.

[Bonus Q: Is the statement also true when 1 and 2 are true over a smaller domain? That is, for example, if the original 1 and 2 equations are only true for 0$\leq y \leq 20$, I think the contradiction still holds.]

[edit: clarified function assumed to be strictly increasing. Removed confusing text to clarify that the general case is different than the specific case]

Answer 1

Investigate only $(1)$ and $(2)$ and only on $\Bbb N$ (or equivalently, $\Bbb N+\delta$). These follow a rerursion for which it is well-known that the solutions are linear of the form $$ \tag1x_n=\sum_i \alpha_i\lambda_i^n=\sum_j\beta_j\mu_j^n$$ where the $\lambda_i$ are the (complex) roots of $f(x)=x^{10}-2x^3+1$ and the $\mu_j$ are the roots of $g(x)=x^{10}-\frac{10}9x^8+\frac19$. Any number of maps of the form $n\mapsto \lambda^n$ with distinct bases are linearly independent. It follows that $(1)$ can only hold if all $\alpha_i$, $\beta_i$ are $=0$ except perhaps those for terms occurring on both sides, i.e., for common roots of $f$ and $g$. The Euclidean algorithm reveals that $\gcd(f(x),g(x))=x-1$, i.e., the only common root of $f,g$ is $1$. We conclude that

$x_n$ is constant.

In other words, any $f\colon[0,\infty)\to \Bbb R$ with $(1)$ and $(2)$ is constant on subsets of the form $\Bbb N+\delta$, i.e., $f$ is periodic with period $1$.

If we add that $f$ is weakly increasing, $f$ must be constant, and if strictly increasing, no such $f$ exists.