Product inside summation in Matlab

Hi, I am trying to code the below series in Matlab. To simplify, I am using a and b here, but essentially they are elemets of vectors.

I wrote it as:

I am not sure how to deal with the product inside the addition. Could anyone help me with how to code this? (a and b are not symbols so symprod would probably not work)

Answer 1

[Torsten]

The expression equals

a^(t+1) + b^(-t*(t+1)/2) * sum_{k=t+1}^{oo} a^(k+1)*b^(k*(k+1)/2)

This should be easy to code.

[Irem Sara]

@Torsten I actually don't think those expressions are equal. It works for k= t+1, but not for any further term in the summation.

[Torsten]

If a and b are constant values, the formula is correct.

If your a and b values depend on t (as might be the case because you write that they are elements of vectors), the formula cannot be used.

[Irem Sara]

@Torsten Correct me if I am wrong but even when a and b are constants, the expansion is

because the power of the b outside the summation does not update.

Do I have a mistake somewhere?

[Torsten]

No, you didn't make a mistake, but

b^((4*t+6)/2) = b^(2*t+3) = b^(t+1)*b^(t+2)

[Irem Sara]

You are completely right, that's my bad! I got confused with the rules of exponents.

Just wondering, if a and b are non constant since they are the elements of a vector, is there a way to code that in? (I don't think I asked the initial question in the best way possible, sorry about the confusion!)

I guess a better way to write this is:

Answer 2

[Torsten]

For infinite vectors (a)_j and (b)_j and oo replaced by Kmax

t = 2;
Kmax = 50;
prod1 = a(t+1:Kmax).^(t+2:Kmax+1);
B = b(t+1:Kmax).^(t+1:Kmax);
prod2 = cumprod(B);
summands = prod1.*prod2;
expr = a(t)^(t+1) + sum(summands)

[Irem Sara]

Great, thank you so much!

[Torsten]

All in one line:

expr = a(t)^(t+1) + sum(a(t+1:Kmax).^(t+2:Kmax+1).*cumprod(b(t+1:Kmax).^(t+1:Kmax)))