Consider the recurrence equation $u_n = f(u_{n-1},\ldots, u_{n-k})\;,$ defined for $n=0,1,2\ldots$.
If this is a linear recurrence and the coefficient function on $u_{n-k}$ is nonzero in $\mathbb{Z}$, then I can solve my recurrence for $u_{n-k}$ and, keeping the same set of initial conditions, define a recursion heading through the negative integers. Is there a name for the operation I have just constructed?
Example: Consider the Fibonacci numbers, starting with $F_0=0,\, F_1=1$, etc. We have $F_{n-2}=F_{n-1} - F_{n}$, and a reindexing gives the sequence $G_n = - G_{n} + G_{n-2}$, defined on $\mathbb{Z}^+$ with initial conditions $G_0 = 0$ and $G_1=1$. Here, a closed form is given by $G_n = (-1)^n F_n$. (It is not difficult to come up with an example in which the connection is non-obvious.)
Are any nice properties preserved under this operation? In particular, I have been looking at holonomic recurrences (in which coefficients of $f$ are rational functions).