# symbolic summation involving kronecker delta

I have to perform symbolically summations of this kind

$\sum_{ijkl} V_{ijkl} c_i c_j c_k \delta_{l,m}$

where $V_{ijkl}$ are quantities which depend on 4 indices and $\delta_{l,m}$ is the kronecker delta. Is it possible in mathematica to perform the summation in order to obtain the symbolical result

$\sum_{ijk} V_{ijkm} c_i c_j c_k$?

I just tried with

Sum[V[ijkl] d, i,j,k,l]


where d contains terms of the kind $c_i c_j c_k \delta_{l,m}$.

• How did you go from the top symbolic expression to the bottom one? What are the limits of summation? What have you tried so far? – MarcoB Oct 1 '15 at 16:10
• Don't use D as your own symbol name; that's a Mathematica built-in for the derivative function. In general, never use uppercase for user-defined symbols, as they could conflict with Mathematica built-ins. – MarcoB Oct 1 '15 at 18:15
• If you want to work extensively with index notation you might want to consider external packages like Ricci: math.washington.edu/~lee/Ricci – Graumagier Nov 2 '15 at 18:39

## 2 Answers

In the documentation of KroneckerDelta says:

Use in sums to pick out elements:

Sum[KroneckerDelta[a, 3] f[a], {a, Infinity}]
(*f*)


Just do the sum in l with {l,Infinity}

Assuming[m > 1 &&
m \[Element] Integers, Sum[
KroneckerDelta[l, m] f[m], {l, Infinity}]]
(* f[m] *)


f is your other sums.

Try it with Sum[V[i j k l] d, i,j,k,l]. In mathematica xy is not the same thing as x y. Mathematica interprets xy as a variable but it interprets x y as x * y, so if you wanted to to V[i*j*k*l] try that. Sorry if this wasn't much help.