Function as a sum over a list of functions

Why this isn't working

I believe your problem is that beta and gamma don't appear on the right hand side (rhs) explicitly in your definition of q, but only through R. This is a problem here because you're using :=, which does not evaluate its rhs until the function being defined is called.

After defining q as you have, evaluate ?q—this will give you the definitions for q, and you'll see that R appears as a symbol there, not as something involving beta and gamma. So, q[Pi/3, Pi/4] gets replaced with Sum[R[[n, 3]], {n, 3}], and Pi/3 and Pi/4 are thrown away. Then R is expanded into its definition.

How to fix it

There are a couple ways to fix this.

1. Make R a function: R[beta_, gamma_] := <insert matrix here>. Then use q[beta_, gamma_] := Sum[R[beta, gamma][[n, 3]], {n, 3}].

2. Use = instead of :=. Unlike :=, = evaluates its rhs first. This is generally riskier and not recommended, since the arguments are not bound; if you have external definitions for the arguments, they'll be substituted in before the definition is created! So for example, x = 3; f[x_] = x + 1 will make f always return 4. This is not the case for f[x_] = x + 1 when x has no prior definition, even if x is defined later, e.g. f[x_] = x + 1; x = 3 produces an f that works as expected. So good variable hygiene is important.

3. Or, like, 2 but safer, inline the whole of R.