Unfortunately, this question has a very bland answer; however, it occurred to me that recursive functions, like ones to program and calculate factorials for instance, are very similar to differential equations whose results depend on initial conditions. Is recursion essentially just discrete DEs, versus the typical continuous DEs learned at school?
Is recursion a type of differential equations
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0I think you are referring to difference equations, they are indeed very similar to differential equations. If you take a differential equation and discretize it in order to solve it numerically you will end up with a difference equations (or recursive equations). – 2017-01-10
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0You might want to check out the Calculus of Finite Differences (https://en.wikipedia.org/wiki/Finite_difference#Calculus_of_finite_differences) - probably a better analogy than recursion. – 2017-01-10
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Discrete Differential equations are called "Difference Equations" and are mostly expressed by $$k_{n+1} - k_{n} = f(n, k) \quad n\in \mathbb{N}$$ These give the difference between two consecutive values.
Recursive Expressions ware mostly stated by $$k_{n+1} = f(n, k)$$ It is always possible to make this into a difference equation by just subtracting $k_n$ on both sides though.
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0If you substract $k_n$ the right hand side ends up being unknown. – 2017-01-10
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0But $f(n,k)$ can already depend on $k_n$. As in $k_{n+1}=k_n +1$. You would need some kind of initial condition, that's true. – 2017-01-10