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Math Help - How do I multiply these sums?

  1. #1
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    How do I multiply these sums?

    I have the following expression, and I want to have a formula that gives me the coefficients of x^{s} of this product.


    (x+x^2+x^3)^2

    This can be written like this:

    x^2(1-x^3)^2(1-x)^{-2}

    Now I used the binomial theorem:
    x^2(1-x^3)^2=\sum_{k=0}^{2}{2\choose k}(-1)^{k}x^{3k+2} and (1-x)^{-2}=\sum_{r=0}^{\infty}{2+r-1\choose r}x^r

    So
    (x+x^2+x^3)^2=\sum_{k=0}^{2}{2\choose k}(-1)^{k}x^{3k+2}\sum_{r=0}^{\infty}{2+r-1\choose r}x^r

    But it doesn't seem right, and I don1t know how to rewrite it.
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  2. #2
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    Re: How do I multiply these sums?

    Is there a reason why you are trying to do it in that convoluted fashion rather than just multiply it?

    ( (x+ x^2+x^3)^2 is NOT equal to x^2(1- x^3)^2(1- x)^{-2}. One rather obvious point is that the first is defined at x= 1 and the second isn't.)
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  3. #3
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    Re: How do I multiply these sums?

    Okay, but if |x|<1 then they are equal aren't they?

    Yes, there is a reason. I want to find a formula that finds these coefficients, for more general cases.
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  4. #4
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    Re: How do I multiply these sums?

    Okay, but still why are you doing that?

    It's simple to see that x(x+ x^2+ x^3)= x^2+ x^3+ x^4,
    x^2(x+ x^2+ x^3)= x^3+ x^4+ x^5, and
    x^3(x+ x^2+ x^3)= x^4+ x^5+ x^6 so that

    ((x+ x^2+ x^3)^2= x^2+ 2x^3+ 3x^4+ 2x^5+ x^6
    Last edited by HallsofIvy; March 23rd 2014 at 06:20 AM.
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  5. #5
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    Re: How do I multiply these sums?

    If I had a formula for this simple case than I would be able to create a similar formul for not so simple cases like.

    (x+x^2+x^3+x^4+x^5+x^6)^m
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  6. #6
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    Re: How do I multiply these sums?

    Quote Originally Posted by gotmejerry View Post
    If I had a formula for this simple case than I would be able to create a similar formul for not so simple cases like.

    (x+x^2+x^3+x^4+x^5+x^6)^m
    LaTeX problems: will try again
    Last edited by JeffM; March 23rd 2014 at 12:39 PM.
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  7. #7
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    Re: How do I multiply these sums?

    Quote Originally Posted by gotmejerry View Post
    If I had a formula for this simple case than I would be able to create a similar formul for not so simple cases like.

    (x+x^2+x^3+x^4+x^5+x^6)^m
    Is this the problem: simplify $\displaystyle \left(\sum_{i=1}^nx^i\right)^m ,\ given\ m,\ n \in \mathbb Z\ and\ m > 0 < n\ and\ x > 0.$

    $Case\ I:\ x = 1 \implies \displaystyle \left(\sum_{i=1}^nx^i\right)^m = \left(\sum_{i=1}^n1^i\right)^m = \left(\sum_{i=1}^n1\right)^m = n^m.$

    $Case\ II:\ x \ne 1.$

    $\dfrac{x^{(n + 1)} - 1}{x - 1} = \displaystyle \sum_{i=0}^nx^i = 1 + \sum_{i=1}^nx^i \implies$

    $\displaystyle \sum_{i=1}^nx^i = \dfrac{x^{(n + 1)} - 1}{x - 1} - 1 = \dfrac{x^{(n + 1)} - 1 - x + 1}{x - 1} = \dfrac{x^{(n + 1)} - x}{x - 1} = \dfrac{x(x^n - 1)}{x - 1}\implies$

    $\displaystyle \left(\sum_{i=1}^nx^i\right)^m = \left(\dfrac{x(x^n - 1)}{x - 1}\right)^m.$
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  8. #8
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    Re: How do I multiply these sums?

    Thanks, but I want to find the coefficients of the powers of x.
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