Insights Reduction of Order For Recursions

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The discussion focuses on the concept of recursion relations, illustrating how they generate sequences like {1, 4, 7, 10} and the Fibonacci series. It highlights the formulation of these sequences using rules, such as a_{n + 1} = a_n + 3 for the first example and F_{n + 2} = F_{n + 1} + F_{n} for Fibonacci numbers. There is a debate over terminology, with participants discussing the use of "recursion relation" versus "recurrence relation" and the preference for "finite difference calculus" over "difference calculus." Some participants express difficulty finding resources on finite difference calculus, indicating a gap in available information. Overall, the conversation emphasizes the foundational understanding of recursion and its terminology in mathematical contexts.
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This is not meant as a full introduction to recursion relations but it should suffice for just about any level of the student.
Most of us remember recursion relations from secondary school. We start with a number, say, 1. Then we add 3. That gives us 4. Now we that number and add 3 again and get 7. And so on. This process creates a series of numbers ##\{ 1, 4, 7, 10, \dots \}##. Once we get beyond that stage we start talking about how to represent these. Well, let’s call the starting number ##a_0##. Then the next number in the series would be ##a_1##, then ##a_2##, …, on to ##a_n## where n is the nth number in the series. We may now express our recursion as a rule: ##a_{n +1} = a_n + 3##, where ##a_0 = 1##.
We may go much further. We can talk about things like the Fibonacci series: ##F_{n + 2} = F_{n + 1} + F_{n}##, where ##F_1 = F_2 = 1##. This is the famous series ##\{ 1, 1, 2, 3, 5, 8, 13, \dots \}##. But what if we want a formula to find out what ##F_n## is for the nth number...

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topsquark said:
Is the term "recursion relation" in common use? I would call it a "recurrence relation".

Also I would say "finite difference calculus" instead of "difference calculus".
 
pbuk said:
Is the term "recursion relation" in common use? I would call it a "recurrence relation".

Also I would say "finite difference calculus" instead of "difference calculus".
Sorry! As I understand it both terms (the recursive and finite difference) are in use. However, the only work I've done with recurrence relations comes from High School and I couldn't really find much on the web about finite difference calculus. I had to fill in a lot of gaps to do this, so I may not be using standard terminology.

-Dan
 
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