Let $(a_n)$ be a sequence of numbers which satisfies the following recurrence relation for all $n\geq 1$,
\[c\cdot a_na_{n-1} + c_1\cdot a_n + c_2\cdot a_{n-1} + c_3=0 \]
where, $c_1, c_2, c_3$ and $c$ are constants, $c\neq 0$, and $a_0\neq-\frac{c_1}{c}$ is given.
Does a general solution in CLOSED FORM to this non-linear recurrence exist ?
*Note: This is an open question. Most non-linear recurrences don't have a solution in closed form. But, I can't seem to find any resource with a treatment of this type of recurrence. Any link/resource/reference/discussion would be helpful...
Solution to a Non-Linear Recurrence
Re: Solution to a Non-Linear Recurrence
Can you provide the problem-specific values? Or is this purely a product of your thought?
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Nur Muhammad Shafiullah | Mahi
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Nur Muhammad Shafiullah | Mahi
Re: Solution to a Non-Linear Recurrence
To tell the truth, it's a product of my thought... I came up with this recurrence while trying to find the Equivalent Resistance of an Infinite Fractal-Circuit that I made up.
The original recurrence I was trying to solve had $c=3, c_1=4, c_2=-2, c_3=-2$ and $a_0=\frac{2}{3}$.
I was wondering if the recurrence could be solved for arbitrary constants (within the given constraints)...
The original recurrence I was trying to solve had $c=3, c_1=4, c_2=-2, c_3=-2$ and $a_0=\frac{2}{3}$.
I was wondering if the recurrence could be solved for arbitrary constants (within the given constraints)...
Re: Solution to a Non-Linear Recurrence
Any help...
Re: Solution to a Non-Linear Recurrence
If it is given(or can be proven in certain cases), that $a_n$ is integer for $n\geq1$, then this recurrence will have finitely many(possibly $0$) solutions.sowmitra wrote:Let $(a_n)$ be a sequence of numbers which satisfies the following recurrence relation for all $n\geq 1$,
\[c\cdot a_na_{n-1} + c_1\cdot a_n + c_2\cdot a_{n-1} + c_3=0 \]
where, $c_1, c_2, c_3$ and $c$ are constants, $c\neq 0$, and $a_0\neq-\frac{c_1}{c}$ is given.
Does a general solution in CLOSED FORM to this non-linear recurrence exist ?
*Note: This is an open question. Most non-linear recurrences don't have a solution in closed form. But, I can't seem to find any resource with a treatment of this type of recurrence. Any link/resource/reference/discussion would be helpful...
Re-write as: $a_n(ca_{n-1}+c_1)=-c_2a_{n-1}-c_3$. So, $ca_{n-1}+c_1|-c_2a_{n-1}-c_3|-c_2ca_{n-1}-c_3c$. From this, and $ca_{n-1}+c_1|c_2ca_{n-1}+c_1c_2$, we have $ca_{n-1}+c_1|c_1c_2-c_3c$. See that, This means, $ca_{n-1}+c_1$ is a divisor of $c_1c_2-c_3c$. Hence, $a_n$ can't assume many distinct values. So, at some point it may be periodic or constant, or whatever. If some more conditions like strictly increasing or other stuff are related, then it may be said that there are no solutions at all.
One one thing is neutral in the universe, that is $0$.