term = 0
for k in 1:N
term_inner = 0
for p in 0:m
term_inner += binomial(m, p) * c[k]^p *f[k]^(m - p) * integral_Ik_recursive(p, γ)
end
term += exp(im * 2π * γ * (k - 1)) * term_inner
end

I want to compute this summation if m and \gamma are vectors. Namely, ı want to fill the array with its dimensional length(m) x length(\gamma) . How can i write effective code?

One option is to put your code into a function and then broadcast it

First, put your code into a function:

function mysum(m, γ; c, f, N)
term = 0
for k in 1:N
term_inner = 0
for p in 0:m
term_inner += binomial(m, p) * c[k]^p *f[k]^(m - p) * integral_Ik_recursive(p, γ)
end
term += exp(im * 2π * γ * (k - 1)) * term_inner
end
return term
end

The arguments to this function are scalars m and γ, but you can apply it to vectors with broadcasting:

mysum.(m, γ'; c, f, N)

Another option is to use broadcasting inside your mysum function so that it works on vectors. This takes a little more work, e.g. initializing term based on the sizes of the vectors, using “dot” operations like @. term += exp(...), and so on.

On a separate note, this is a very inefficient way to sum a series like this. You are recomputing the binomial factor and powers like c[k]^p separately for each loop iteration, when in practice you can compute them much more efficiently using a recurrence relation from the previous term.