From my textbook:
"A recurrence relation of degree k is defined by a function $p: N \times Z^k \rightarrow Z $ and an initial sequence of integers $a_0,a_1,...,a_k$ such that for all $n\ge k$ the number $a_n$ is equal to $p(n,a_{n-1},...,a_{n-k})$. The problem with this definition is that it describes any function on the natural number if there is no restriction on the properties of the function p. But usually some form of restriction on p is implicit in this definition."
My questions:
So the function $p$ has the domain of paired natural numbers and k-dimensional integers?and it maps to the set of integers? Does it mean when the inputs are $n,a_{n-1},...,a_{n-k}$ then it maps to $a_n$?
If the sequence is a function from $N$ to $R$, so in the definition $Z$ should be replaced by $R$?
- Why does p need to have some form of restriction and what is the problem that it describes any function on the natural numbers?