Timeline for Linear two-dimensional recurrence relation
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Jun 16, 2020 at 13:19 | vote | accept | Carlos Camino | ||
| Apr 18, 2017 at 0:43 | answer | added | user44191 | timeline score: 1 | |
| Apr 10, 2017 at 8:19 | comment | added | Carlos Camino | Thank you again for your answers. Unfortunately I do not understand your last post. Could you please be more specific? From $f'_{m,n} = f_{m,n} + \frac{1}{1-a-b}$ and $f'_{m,n} = af_{m-1,n} + bf_{m,n-1}$ it follows $c = - \frac{1}{1-a-b}$, which is not necessarily right. What am I missing? | |
| Apr 7, 2017 at 20:01 | comment | added | user44191 | You can change that first double sum into two single sums by considering $f'_{m,n} = f_{m,n} + \frac{1}{1-a-b}$, which has the recurrence relation $f'_{m,n} = a f_{m-1,n} + b f_{m,n-1}$, which results in a formula with no double sum. | |
| Apr 7, 2017 at 18:13 | comment | added | user44191 | $\sum_{j=0}^{n-1} {{i+j}\choose j} b^j$ is the first $n$ terms of $\frac{1}{1-b}^i$; if there is a nice formula for $\sum_{j=n}^\infty {{i+j}\choose i} b^j$, you should be done. | |
| Apr 7, 2017 at 13:53 | comment | added | Carlos Camino | You are right, thank you! The first term is the one I am trying to simplify right now. | |
| Apr 7, 2017 at 13:41 | history | edited | Carlos Camino | CC BY-SA 3.0 |
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| Apr 6, 2017 at 12:42 | comment | added | Lwins | It seems that we could hardly simplify the form of your result because that evidently each $f_{0,j}\ (j \leq m)$ and $f_{i,0}\ (i \leq n)$ has to appear at least once in the formula. Maybe we could simplify the first term $c \sum \sum \binom{i+j}{i} a^i b^j$? | |
| Apr 6, 2017 at 9:44 | history | edited | Carlos Camino | CC BY-SA 3.0 |
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| Apr 6, 2017 at 9:00 | comment | added | Carlos Camino | Perhaps "closed-form solution" is not the right term here. I know from one-dimensional linear recurrence relations (like $f_n = a f_{n-1} + b f_{n-2}$) that their solutions are very simple. I was hoping to be able to make it nicer. Maybe getting rid of the sum signs. | |
| Apr 6, 2017 at 0:59 | answer | added | T. Amdeberhan | timeline score: 6 | |
| Apr 5, 2017 at 17:24 | comment | added | user44191 | That seems about as closed-form as it can be; it uses the minimal amount of information you can get, and seems to combine terms wherever possible. What more would you be looking for for "closed form"? | |
| Apr 5, 2017 at 16:33 | review | First posts | |||
| Apr 5, 2017 at 16:37 | |||||
| Apr 5, 2017 at 16:32 | history | asked | Carlos Camino | CC BY-SA 3.0 |