fog37 said:
Interesting. I didn't know that.
Ok, but you didn't answer the question. From reading your posts, I gather that you have only studied mathematics from an informal and intuitive point of view. That's where everyone starts out. Answering the question precisely requires understanding math rigorously. You may not be ready to do that.
Just to paraphrase, so I understand clearly, let's keep considering the function like ##f(x) = x^2##. The dependent values of this function satisfies the specific recursive equation ##f(x) = 2f(x-1) - f(x-2) + 2## with the initial condition ##f(1) = 1##.
If we changed the initial condition, that specific recursive equation would not work anymore, correct?
If by "not work", you mean that ##f(x) = x^2## might no longer be a solution then, yes, that is correct. However some conditions like ##f(2) = 4## still allow ##f(x) = x^2## to be a solution.
In this case, the recursive equation starts at positive ##x=1## (the seed) and provides the values of ##f(x)## for subsequent integer values of ##x##.
Often, when you compute values for a function ##f(x)## using a recursive equation, you began the calculation at ##x = 2## having been given the value of ##f## at ##x = 1##. This is because recursive equations are often (but not always) meant to apply only for positive integer values of ##x##. Is that what you mean? The equation itself doesn't "start" at a particular value of ##x##.
The equation ##f(x) = x^2##, on the other hand, exists for both continuous and discrete positive and negative values of ##x##.
It is possible to write recursive equations with the understanding that the argument of the function can be any real number. The person writing the equation should make it clear if that is the case. Sometimes the domain of a function is implied by the general context where it appears. There is no hard and fast rule that a recursive equation applies only to discrete or integer values.
Could a different "initial condition+recursive equation" represent the same function
##f(x) = x^2##? I would think so...
You have to define what you mean by "represent". To do that, you must first understand why the informal use of the word "represent" is ambiguous. Equations can have several different solutions.
My sense is that if we have an equation ##f(x)##, we can always find a specific pair [recursive equation + initial condition] that matches the the values generated by ##f(x)## starting from ##x=1##..
It's unclear what you mean by "have an equation". If we set ##f(x)## equal to the outcome of a computer algorithm whose input is ##x##, do we "have an equation" for it? A computer algorithm can compute the terms for the sequence given in post #13.
Also, the initial condition is not limited to start at ##x=1##, correct? We could have ##f(-2)= initial value##, correct?
Yes, it's correct that an "initial condition" might ( or might not) be given for values of a function when the argument is -2, or other numbers different than 1. It depends on the full context of where the equation appears.
Trying to deduce the meaning of a single equation without considering the context where it appears is like trying to interpret the sentence "It is better than the other thing" without seeing the context where it is written.