Hello, I'm working with a system of equations that has an infinite recursion function, and am wondering if its possible to simplify or remove the recursion in terms of the other functions in the system. Any insight into the framework or family of this system is appreciated.
Given two functions P(x) and C(x), define a 3rd function A(x), such that:
A(x) = 2x - P(x) - C(x)
Then define an infinitely recursive composite 4th function Q(x), such that:
Q(x) = ...A(x) +C(A(x)+C(A(x)+C(A(x))))
Can Q(x) be simplified/solved to remove the recursion and/or shown in terms of a non recursive function of P(x) and C(x)?
I did notice that it's almost stating that Q(x) = A(x) +C(Q(x)), and was going to try and approach it that way, but that's not quite right, seeing as the recursion is on the front end and not terminating on the inside. Or does that represent it correctly? In which case how do I get the Q(x) out from the input of C(x)? Is it solely dependent on the form of C(x) itself, or removable by some other method?
Also, I was able to think of Q(x) as a series of equations, Qn(x), such that:
Q1(x) = C(A(x))
Q2(x) = C(A(x)+Q1(x))
Q3(x) = C(A(x)+Q2(x)) ... or in general,
Qn(x) = C(A(x)+Qn-1(x))
in which case, the infinite recursion is represented as Q∞(x), but I didn't really know where to go from here.
Anyways, thanks for any insight into this system.