how to generate functions with changing array size?

Question

I want to write a function in Mathematica which has two inputs 'n' and 'd', and the output should be (in Latex notation)

$$F(n,d) = \sum_{i[1],...,i[n] =0; i[1]+...+i[n]\leq d}^{d} C[i[1],...,i[n]] f[i[1],x[1]] f[i[2],x[2]] ...f[i[n],x[n]]$$

n and d are integers.

The problem is that I can hard code F for a given n and d alright. But when I want to write a generic function which take in any n and d and spits out the required right hand side, it becomes difficult as I have to keep changing the array size along the way depending on the values of n and d. Probably, there is a simpler way to do this in Mathematica?

Answer 1

The f[x[i]] factor out of the problem, so the only problem is to determine the range of the sum. Here is one way (naming the function makeSum instead of F).

Edit:

The summation range initially didn't include 0, but this can be specified as a third argument to IntegerPartitions:

makeSum[n_, d_] := 
 Sum[C @@ i Product[f[i[[j]], x[j]], {j, n}], {i, 
   Join @@ Table[IntegerPartitions[i, {n}, Range[0, d]], {i, 0, d}]}]

makeSum[2, 5]

(*  ==>
C[0, 0] f[0, x[1]] f[0, x[2]] + C[1, 0] f[0, x[2]] f[1, x[1]] + 
 C[1, 1] f[1, x[1]] f[1, x[2]] + C[2, 0] f[0, x[2]] f[2, x[1]] + 
 C[2, 1] f[1, x[2]] f[2, x[1]] + C[2, 2] f[2, x[1]] f[2, x[2]] + 
 C[3, 0] f[0, x[2]] f[3, x[1]] + C[3, 1] f[1, x[2]] f[3, x[1]] + 
 C[3, 2] f[2, x[2]] f[3, x[1]] + C[4, 0] f[0, x[2]] f[4, x[1]] + 
 C[4, 1] f[1, x[2]] f[4, x[1]] + C[5, 0] f[0, x[2]] f[5, x[1]]    *)

In using IntegerPartitions, I assumed that you want to avoid double-counting. This may have to be modified if the condition includes permutations:

makeSum[n_, d_] := 
 Sum[C @@ i Product[f[i[[j]], x[j]], {j, n}], {i, 
   Flatten[Permutations /@ 
     Join @@ Table[IntegerPartitions[i, {n}, Range[0, d]], {i, 0, d}],
     1]}]

makeSum[2, 5]

(*
==> 
C[0, 0] f[0, x[1]] f[0, x[2]] + C[1, 0] f[0, x[2]] f[1, x[1]] + 
 C[0, 1] f[0, x[1]] f[1, x[2]] + C[1, 1] f[1, x[1]] f[1, x[2]] + 
 C[2, 0] f[0, x[2]] f[2, x[1]] + C[2, 1] f[1, x[2]] f[2, x[1]] + 
 C[0, 2] f[0, x[1]] f[2, x[2]] + C[1, 2] f[1, x[1]] f[2, x[2]] + 
 C[2, 2] f[2, x[1]] f[2, x[2]] + C[3, 0] f[0, x[2]] f[3, x[1]] + 
 C[3, 1] f[1, x[2]] f[3, x[1]] + C[3, 2] f[2, x[2]] f[3, x[1]] + 
 C[0, 3] f[0, x[1]] f[3, x[2]] + C[1, 3] f[1, x[1]] f[3, x[2]] + 
 C[2, 3] f[2, x[1]] f[3, x[2]] + C[4, 0] f[0, x[2]] f[4, x[1]] + 
 C[4, 1] f[1, x[2]] f[4, x[1]] + C[0, 4] f[0, x[1]] f[4, x[2]] + 
 C[1, 4] f[1, x[1]] f[4, x[2]] + C[5, 0] f[0, x[2]] f[5, x[1]] + 
 C[0, 5] f[0, x[1]] f[5, x[2]]
*)