2D variable coefficient recurrence relation

In summary, the conversation discusses the complexity of finding explicit formulas for 2D variable coefficient linear recurrence relations. The example provided has a solution involving factorials. The speakers also mention the difficulty of finding nice formulas for difference relations compared to differential equations, and inquire about the existence of an algorithm for non-constant coefficient 2D linear recurrence relations.
  • #1
rsq_a
107
1
Consider a 2D variable coefficient linear recurrence relation. An example might be:

[tex]b_{n,j+1} (j+1)(2n-1)(2n-2) = (2n-2+j)(2n-1+j)b_{n-1,j}[/tex]

which has the solution
[tex]b_{n,j} = \frac{(2n-1+j)!}{(2n-1)!j!}[/tex]

Is there any algorithm that can be used to derive this result? I have a recurrence relation which is a bit more complex than this one.
 
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  • #2
Difference relations tend to be UGLY with respect to finding nice, explicit formulae.
The continuous analogue, diff. eqs, tend to be easier to generalize about.
 
  • #3
arildno said:
Difference relations tend to be UGLY with respect to finding nice, explicit formulae.
The continuous analogue, diff. eqs, tend to be easier to generalize about.

I realize that. I know that for constant coefficient 2D linear recurrence relations, it seems that Z-Transforms can be applied and are mostly successful. I was wondering whether anybody had seen an analogous technique for non-constant coefficient ones.
 

1. What is a 2D variable coefficient recurrence relation?

A 2D variable coefficient recurrence relation is a mathematical equation that describes a sequence of numbers in two dimensions, where each term is defined in terms of the previous terms and coefficients can vary based on the position in the sequence. It is commonly used in areas such as computer graphics and image processing.

2. How is a 2D variable coefficient recurrence relation different from a regular recurrence relation?

A regular recurrence relation only has one dimension, while a 2D variable coefficient recurrence relation has two dimensions. This means that the terms in the sequence are defined in terms of two previous terms instead of just one. Additionally, the coefficients in a 2D recurrence relation can vary based on the position in the sequence, while in a regular recurrence relation they are usually constant.

3. What is the purpose of using a 2D variable coefficient recurrence relation?

A 2D variable coefficient recurrence relation is often used to model real-world phenomena that have two-dimensional patterns, such as images or signals. It can also be used to efficiently calculate values in a sequence, as it only requires knowledge of the previous two terms instead of all previous terms.

4. How is a 2D variable coefficient recurrence relation solved?

To solve a 2D variable coefficient recurrence relation, various methods can be used such as substitution or matrix algebra. The specific method used will depend on the specific form of the equation and the coefficients involved.

5. What are some applications of 2D variable coefficient recurrence relations in science?

2D variable coefficient recurrence relations are used in many areas of science, including computer science, physics, and engineering. They are commonly used in image and signal processing, numerical analysis, and data compression. They can also be applied to model natural phenomena such as population growth or chemical reactions.

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