How to replace the symbol n in DifferenceRoot

Question

I'd like to employ another symbol (say alpha or k) instead of n in the following (type of) function:

DifferenceRoot[
  Function[{\[FormalY], \[FormalN]},
    {2 + 3 \[FormalN] - 2 \[FormalN]^2 - \[FormalN]^3 - (1 + \[FormalN]) (2 + \[FormalN])
     \[FormalY][\[FormalN]] + \[FormalN] (1 + \[FormalN]) (2 + \[FormalN])
     \[FormalY][1 + \[FormalN]] == 0, \[FormalY][1] == 0}
   ]
 ][n]

Answer 1

There is a hidden option to change the variables used by DifferenceRoot[]:

SetSystemOptions["HolonomicOptions" -> {"IteratorSymbol" -> k, "SequenceSymbol" -> ϕ}];

after which

DifferenceRootReduce[n!, n]
   (* DifferenceRoot[Function[{ϕ, k}, {(-1 - k) ϕ[k] + ϕ[1 + k] == 0, ϕ[0] == 1}]][n] *)

For the analogous case of DifferentialRoot[], the corresponding options are "VariableSymbol" and "FunctionSymbol".