Can one give a general formula for the product of three different finite sums? See the picture attached with this question.
Dear DR Feng and Qi and Viera Cernanova
Thanks for your efforts and time on this problem.
I would like to invite you, kindly join me and let we start writing details and prepare an research paper on it.
Wishes
Right.
My first solution holds in the case when the machine doesn’t know to stop, if there are no more terms to add. The second holds if the machine stops automatically after having passed through all terms (i.e. sum up to k=l, ..., stop).
A formula convenient to your wishes could be constructed e.g. introducing characteristic functions Xk(i,j).
Use the Kronecker Delta as may be seen in the attached picture in an "by hand" approach. (Please do not mix up the summation index e with the upper summation limit l !) You can use it to reduce the coefficient you are looking for to three different representations by a double sum in the obvious way. No conditions on l,m,n are needed. The approach can also be used for (formal) power series.
Let me add a comment: This approach based on Kronecker delta is part of first year courses "mathematics for physicists", at least it was, when I studied. It is a general tool useful also for much more complicated expressions and belongs into the toolbox of everybody using mathematics in research.
Of course, the corresponding formula (the polynomial expansion) given by Viera Čerňanová may be generalized to any n different (polynomial) finite sums of this type, where n is an arbitrary positive integer gretaer than one.
The previously mentioned generalized formula for the product of any n different finite sums can be proved by mathematical induction on n > 1 (with the induction base n = 2).
- Badges
- Science topic
- Similar topics
- Mathematical Sciences