Changing the summation indexes in double sums.

In summary, the variable switch made is a valid one as both sides of the equation are polynomials in x with matching coefficients. To prove this, it can be shown that the coefficient of xh on the right side is equal to the coefficient of xh on the left side. This is done by using the fact that \binom{m}{h-i} is zero when i > h, making both sums equal to \sum_{i=0} ^h \binom{n}{i}\binom{m}{h-i}.
  • #1
quasar987
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I have just made the following variable switch:

[tex]\sum_{i=0}^n\sum_{j=0}^m\binom{n}{i}\binom{m}{ j}x^{i+j}=\sum_{k=0}^{n+m}\sum_{i=0}^k\binom{n}{i}\binom{m}{k-i}x^{k}[/tex]

I know it's right, but is there a method I can use to prove without a shadow of a doubt that it is?
 
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  • #2
They're both polynomials in x, just match up coefficients. For 0 < h < n+m, the coefficient of xh on the right is:

[tex]\sum _{i=0} ^h \binom{n}{i}\binom{m}{h-i}[/tex]

On the left, it's:

[tex]\sum _{i=0} ^n \binom{n}{i}\binom{m}{h-i}[/tex]

but [itex]\binom{m}{h-i}[/itex] is zero when i > h, so the above is really equal to:

[tex]\sum _{i=0} ^h \binom{n}{i}\binom{m}{h-i}[/tex]

which we've already seen to be the coefficient on the left side.
 

1. What is the purpose of changing the summation indexes in double sums?

The purpose of changing the summation indexes in double sums is to simplify the calculation process and make it easier to understand. By changing the indexes, the summation can be rearranged in a way that makes it easier to manipulate and solve.

2. How do I know when to change the summation indexes in double sums?

You can change the summation indexes in double sums when you want to express the same sum in a different form. This can be helpful when trying to simplify the calculation or when trying to find a specific pattern in the summation.

3. Can changing the summation indexes affect the final result of the double sum?

No, changing the summation indexes does not affect the final result of the double sum. It only changes the way the sum is expressed and does not change the actual calculation. As long as the new indexes follow the same rules as the original indexes, the result will remain the same.

4. Are there any specific rules or guidelines for changing the summation indexes in double sums?

Yes, there are a few rules to keep in mind when changing the summation indexes in double sums. The new indexes should still cover the same range as the original indexes, and they should also follow the same pattern of increment or decrement. Additionally, the new indexes should not overlap or skip any values in the summation.

5. Can changing the summation indexes make the calculation process more complicated?

In some cases, changing the summation indexes can make the calculation process more complicated. This is why it is important to carefully consider the changes and make sure they do not create any errors or inconsistencies in the calculation. It is always recommended to double check the results after changing the summation indexes.

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