Sorry, I am having trouble relating your code to your equation, and I would have no idea how to do this with syms. But maybe good old fminsearch will help, something like this:
m = rand(10,1)*3;
t = rand(10,1)*3;
startXs = [1 1 1 1 1];
fun2Minimize = @(x) dblSumFun(x(1),x(2),x(3),x(4),x(5),m,t);
ests = fminsearch(fun2Minimize,startXs)
function dblSum = dblSumFun(u,k,alpha,c,P,m,t)
dblSum = 0;
mc = 2.5;
jSum = 0;
jTerm = k * exp( alpha*(m(j) - mc) ) / (t(i) - t(j) + c)^P;
jSum = jSum + jTerm;
iTerm = log(u + jSum);
dblSum = dblSum + iTerm;
Note that I have ignored the bounds that you specified. Maybe you won't need those if the best parameters are inside the boundaries. If you do need them, a little more code will be required inside dblSumFun. But I don't want to elaborate because maybe this is not at all what you want.
By the way, don't start the parameter values near the boundary as you seem to have done in your sample code. Start at the most plausible values you can think of, which are presumably more in the middle of the range.