Defining a function of arguments with sub- and superscripts

Question

I want to define a function that takes, several, let's say two arguments, both of which have several sub- and suberscripts. For example a function might have the signature $$ f(x_i,q_{j,k}) = \mathrm{function\ of\ } x_i, q_{j,k} $$ Now I want the notebook to be so flexible as to allow for $f$ to take arbitrary arguments with abitrary values of indices, e.g: $$ f(a,b) = \mathrm{function\ of\ } a,b $$ $$ f(x_{i+5},q_{j+1,k}) = \mathrm{function\ of\ } x_{i+5}, q_{j+1,k} $$ that is, I want the value of f to change whether I change the values passed or the indices. Of course, in the former case, if the value of f depends explicitly on the value of one of the indices of one of the parameters, say $$f(x_i)=x_i*i$$ then $$f(a)$$ would not necessarily a properly defined expression, unless $a$ can be written as some $x_j$.

Is there a way to define such functions?

Answer 1

Following the comments I am encouraging the use of brackets rather than subscripts or superscripts. Here is an example where a function may take a variable with a subscript or a variable without a subscript. The function will use pattern recognition to sort out how to behave. First we define the function. Note I start with a ClearAll[f] so that previous versions of f are deleted

ClearAll[f];
f[x_[i_]] := 5 x[i] + i;
f[a_] := 2 a 

Now we can try this out.

f[z[3]]

gives

3 + 5 z[3]

while

f[c]

gives

2 c

Is this along the lines of what you are seeking?

Edit

At the request of a comment here is an example where one function has several different outputs which depend on the type of input.

ClearAll[f];
f[x_[i_]] := { 5 x[i] + i, "Indexed Variables"};
f[a_Symbol] := { 10 a , "Symbol"};
f[a_Integer] := { 20 a, "Integer"};
f[a_Real] := {30 a, "Real"};
f[a_] := {40 a , "Anything Else"};

Here is a list of different inputs that we feed into the function

tests = {y[i], z[3], s, 3, 4., a + b^2};
results = f[#] & /@ tests;
TableForm[results]

With output

i+5 y[i]    Indexed Variables
3+5 z[3]    Indexed Variables
10 s    Symbol
60  Integer
120.    Real
40 (a+b^2)  Anything Else

This approach is standard in Mathematica. I am sure there are good examples elsewhere but I could not find them. If you know one please add the link to this answer.

Answer 2

Below I have defined a few different forms which will do exactly as you asked. While I, too, agree with the sentiments iterated by user @Hugh regarding the use of sub- and superscripts, that this is possible with the Wolfram Language must be noted and demonstrated. Regardless, please see the following:

ClearAll[f]; f[Subscript[x_, i_]] := Subscript[x, i] i; f[x_^i__] := x^i/i; f[x_] := x; f[Subscript[x_, i__], Subscript[y_, j__]] := g[Subscript[x, i], Subscript[y, j]]; f[x_, y_] := x + y;

Where g is a function of your definition. Now, we can test this out:

f[Subscript[x, i], Subscript[q, j, k]]
(* g[Subscript[x, i], Subscript[q, j, k]] *)
f[a, b]
(* a + b *)
f[Subscript[x, i + 5], Subscript[q, j + 1, k]]
(* g[Subscript[x, 5 + i], Subscript[q, 1 + j, k]] *)
f[Subscript[x, i]]
(* i Subscript[x, i] *)
f[n^j]
(* n^j/j *)