@Carl Love Using Preben's code, I now understand why f(1) is -x (actually x-> -x) and f(x) is -1. What was confusing me was that I thought that P(x) was always 1+x+x^2.

@Alejandro Jakubi While we cannot construct a patterned matrix of unspecified (abstract) size in Maple, all the OP wants is a procedure that constructs it for specified sizes. This is quite doable in Maple.

@xcyborg You can change the options on this plot, of course. I just made some stuff up.

plot(
     [zip(`[]`, W, M), zip(`[]`, W, B)],
     style= point, symbol= [diamond, box],
     symbolsize= 15, color= [brown, green]
);

@mahmood180 

Replace the G2 loops by the single statement:

G2:= Matrix(r, (i,j)-> `if`(i+j <= r, z[i+j+r+1], 0));

When you use a[k]^`*`, do you mean the complex conjugate of a[k]? Or is it just a notation to indicate something different from a[k]?

You say that the expected result of f(1) is -x and that the expected result of f(x) is -1. Neither of those make sense to me. Could you explain why you expect those results? Then maybe I can help you with the rest. Writing a function within a function is easy.

I just need a mathematical explanation; no knowledge of or use of Maple is needed.

@ecterrab Can you provide an explicit example showing the difference between PDEtools:-Solve and solve to which you are referring? I get identical RootOf results from both PDEtools:-Solve(Theta=F(s), s) and solve(Theta=F(s), s).

@mahmood180

You need to correct the Error, Vector index out of range. Vector z has 2*r+1 elements, but the code z[i+j+r+1] is trying to access z[2*r+2] (because i = r-1 and j = 2).

@Markiyan Hirnyk Your Maple output from the latex command is different from mine. Mine does not have any internal square brackets, and mine uses underscores to indicate subscripting.

Please give a more-detailed example.

@casperyc add is the correct thing to use; sum is incorrect, or at least suspicious. Quoting from ?sum :

To add a finite sequence of values, rather than compute a formula, use the add command.  For example, add(k, k=0..9) returns 45.  Although the sum command can often be used to compute explicit sums, it is strongly recommended that the add command be used in programs if an explicit sum is needed, in particular, when summing over all elements of a list, Array, Matrix, or similar data structure.

@NYK Ah, what exactly do you mean by "type in a subspace"? The canonical representation of a subspace is a basis. Most subspaces cannot be represented by a single (scalar) equation. If you want to work directly with the equation, there's this:

Student:-LinearAlgebra:-NullSpace(
     Student:-LinearAlgebra:-GenerateMatrix(
          [2*x-3*y+6*z], [x,y,z]
     )[..,1..-2]
);

@Chia Matrices can use a very large amount of memory. Making a copy of a Matrix necessarily doubles that memory usage. In-place operation avoids that.

@Chia The print commands print the four numbers. The output of the map command is the last line only.

@Chia Since an expression sequence cannot contain NULLs, the four NULLs which are the outputs of the print commands collapse to a single NULL expression sequence. When this expression sequence is turned into a list, it is the empty list, represented as [].

@Chia I am not getting your point. What is the inconsistency? Note that the numbers printed are the side-effect results of the four print commands, and the Vector of NULLs is the output of the map command. Also note that a Vector can contain NULL entries but an expression sequence cannot. The expression sequence can be NULL, but it cannot contain NULLs.